// Numbas version: exam_results_page_options
{"name": "Matrices revision", "metadata": {"description": "A collection of questions on solving equations and revising manipulation of small matrices for 2nd year students&nbsp;", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "80", "showQuestionGroupNames": true, "showstudentname": true, "question_groups": [{"name": "Revision test for 2nd Year", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "John's copy of Gaussian elimination to solve a 3x3 system of linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Praneetha Singh", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2552/"}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "statement": "<p>Solve the system of equations using Gauss Elimination <br/>\\[\\begin{eqnarray*} &amp;\\var{a}x&amp;+\\;&amp;\\var{a*b-1}y&amp;+\\;\\var{a^2*b-a-a*b}z&amp;=&amp;\\var{c2}\\\\ &amp;\\var{a*c}x&amp;+\\;&amp;\\var{c*b}y&amp;+\\;z&amp;=&amp;\\var{c1}\\\\ &amp;x&amp;+\\;&amp;\\var{b}y&amp;+\\;\\var{b*a-b}z&amp;=&amp;\\var{c3} \\end{eqnarray*} \\] <br/><strong>Part a)</strong> Rearrange the order of the equations and represent this as a system of equations using a matrix.<br/><strong>Part b)</strong> Introduce zeros in the first column using the first row.<br/><strong>Part c)</strong> Introduce zeros in the second <g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\"><g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\"><g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\"><g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"10\" data-gr-id=\"10\">coumn</g></g></g></g> below the second entry in the second row using the second row. <br/><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\"><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\"><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\"><g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\">Also</g></g></g></g> need to solve for $z$ using the last row of the reduced matrix.<br/><strong>Part d)</strong> Solve for $y$ and $x$ using the second and first rows of the reduced matrix.</p>", "parts": [{"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "\n        <p>Re-arrange the rows so that the third row becomes the first row, the first the second and the second the third.<br/><strong>WHY?</strong> Choose one of the following:<br/>[[0]]</p>\n      <p>Now write down the entries of the matrix you will use for Gaussian Elimination, remember to include the constants as the last column.</p>\n      <table>\n      <tbody>\n      <tr>\n      <td rowspan=\"3\">\\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      <td>[[1]]</td>\n      <td>[[2]]</td>\n      <td>[[3]]</td>\n      <td>[[4]]</td>\n      <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      </tr>\n      <tr>\n      <td>[[5]]</td>\n      <td>[[6]]</td>\n      <td>[[7]]</td>\n      <td>[[8]]</td>\n      </tr>\n      <tr>\n      <td>[[9]]</td>\n      <td>[[10]]</td>\n      <td>[[11]]</td>\n      <td>[[12]]</td>\n      </tr>\n      </tbody>\n      </table>\n        ", "gaps": [{"matrix": [1, 0, 0, 0], "scripts": {}, "shuffleChoices": true, "customName": "", "showCorrectAnswer": true, "displayType": "radiogroup", "maxMarks": 1, "choices": ["<p>To make sure that there is a 1 or -1 in the first row, first column position because that makes the arithmetic easier.</p>", "<p>Because you always do this swap.</p>", "<p>To make sure that there is a 1 or -1 in the first row, first column position because that makes the arithmetic more stable.</p>", "<p>I don't know.</p>"], "useCustomName": false, "showCellAnswerState": true, "displayColumns": 0, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "marks": 0, "unitTests": [], "variableReplacements": [], "distractors": ["", "", "", ""], "minMarks": 0, "type": "1_n_2", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst"}, {"allowFractions": true, "maxValue": "1", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "1", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{b}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{b}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{b*a-b}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{b*a-b}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{c3}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{c3}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{a}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{a}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{a*b-1}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{a*b-1}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{a^2*b-a-a*b}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{a^2*b-a-a*b}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{c2}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{c2}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{a*c}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{a*c}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{b*c}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{b*c}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "1", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "1", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{c1}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{c1}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}, {"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "\n      \n      \n      <p>Now introduce zeros in the first column below the first entry by adding:<br/>[[0]] times the first row to the second row and<br/>[[1]] times the first row to the third row to get the matrix:</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\"> \\[\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> $\\var{b}$ </td>\n      \n      <td> $\\var{b*a-b}$ </td>\n      \n      <td> $\\var{c3}$ </td>\n      \n      <td rowspan=\"3\"> \\[\\left) \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$ </td>\n      \n      <td> [[2]] </td>\n      \n      <td> [[3]] </td>\n      \n      <td> [[4]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> [[5]] </td>\n      \n      <td> [[6]] </td>\n      \n      <td> [[7]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>Next multiply the second row by [[8]] to get a 1 in the second entry in the second row.</p>\n      \n      \n        ", "gaps": [{"allowFractions": true, "maxValue": "{-a}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{-a}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{-a*c}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{-a*c}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{-1}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{-1}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.6, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{-a}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{-a}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.6, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{c2-a*c3}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{c2-a*c3}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.6, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{c*b-c*b*a}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{c*b-c*b*a}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.6, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{-a^2*b*c+1+a*b*c}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{-a^2*b*c+1+a*b*c}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.6, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{c1-a*c*c3}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{c1-a*c*c3}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.6, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{-1}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{-1}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}, {"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "\n      \n      \n      <p>Note that you should have multiplied the second row by a suitable number to get a $1$ in the second entry in the second row.<br/>In this part we introduce a $0$ in the second column below the second entry in the second column by adding:<br/>[[0]] times the second row to the third row to get the matrix:</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\"> \\[\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> $\\var{b}$ </td>\n      \n      <td> $\\var{b*a-b}$ </td>\n      \n      <td> $\\var{c3}$ </td>\n      \n      <td rowspan=\"3\"> \\[\\left) \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> $\\var{1}$ </td>\n      \n      <td> [[1]] </td>\n      \n      <td> [[2]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> $\\var{0}$  </td>\n      \n      <td> [[3]] </td>\n      \n      <td> [[4]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>From this you should find:</p>\n      \n      \n      \n      <p>$z=\\;\\;$[[5]]</p>\n      \n      \n        ", "gaps": [{"allowFractions": false, "maxValue": "{-b*c+a*b*c}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{-b*c+a*b*c}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{a}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{a}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{a*c3-c2}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{a*c3-c2}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{1}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{1}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{b*c*(1-a)*(c2-a*c3)+c1-a*c*c3}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 0.8, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": true, "maxValue": "{z}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{z}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 2, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}, {"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "\n      \n      \n      <p>From the second row of the reduced matrix you find an equation involving only $y$ and $z$ and using your value for $z$ we find:</p>\n      \n      \n      \n      <p>$y=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>Then using the first row we have the equation :<br/>\\[\\simplify[all]{x+ {b}y+{b*a-b}z={c3}}\\]</p>\n      \n      \n      \n      <p>Using this you can now find $x$:</p>\n      \n      \n      \n      <p>$x=\\;\\;$[[1]]</p>\n      \n      \n        ", "gaps": [{"allowFractions": false, "maxValue": "{y}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{y}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 3, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}, {"allowFractions": false, "maxValue": "{x}", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "minValue": "{x}", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "useCustomName": false, "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 2.6, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "customName": ""}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "c", "b", "c3", "c2", "y", "x", "c1", "z"], "variables": {"z": {"name": "z", "templateType": "anything", "group": "Ungrouped variables", "definition": "c1+c2*c*(b-a*b)+c3*c*(a^2*b-a-a*b)", "description": ""}, "c1": {"name": "c1", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": ""}, "b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": ""}, "c": {"name": "c", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": ""}, "c2": {"name": "c2", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": ""}, "c3": {"name": "c3", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": ""}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": ""}, "y": {"name": "y", "templateType": "anything", "group": "Ungrouped variables", "definition": "a*c3-c2-a*z", "description": ""}, "x": {"name": "x", "templateType": "anything", "group": "Ungrouped variables", "definition": "c3-(b*a-b)*z-b*y", "description": ""}}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>Look at the revealed answers for this question. All the information needed is there.</p>", "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.</p>"}, "type": "question"}, {"name": "John's copy of Matrix arithmetic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}], "ungrouped_variables": ["a11", "a12", "a21", "a22", "k1", "k2", "b11", "b12", "b21", "b22"], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Given the matrix:</p>\n<p style=\"padding-left: 30px;\">\\(A=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "preamble": {"js": "", "css": ""}, "functions": {}, "rulesets": {}, "variables": {"b22": {"name": "b22", "templateType": "anything", "group": "Ungrouped variables", "definition": "{a21}*{a12}+{a22}^2+{k1}*{a22}+{k2}", "description": ""}, "k2": {"name": "k2", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(6 .. 12#1)", "description": ""}, "b21": {"name": "b21", "templateType": "anything", "group": "Ungrouped variables", "definition": "{a12}*{a11}+{a22}*{a12}+{k1}*{a12}", "description": ""}, "b11": {"name": "b11", "templateType": "anything", "group": "Ungrouped variables", "definition": "{a11}^2+{a12}*{a21}+{k1}*{a11}+{k2}", "description": ""}, "a21": {"name": "a21", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(2 .. 9#1)", "description": ""}, "b12": {"name": "b12", "templateType": "anything", "group": "Ungrouped variables", "definition": "{a21}*{a11}+{a22}*{a21}+{k1}*{a21}", "description": ""}, "a22": {"name": "a22", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(11 .. 21#1)", "description": ""}, "a11": {"name": "a11", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(1 .. 10#1)", "description": ""}, "a12": {"name": "a12", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(0 .. 10#1)", "description": ""}, "k1": {"name": "k1", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(2 .. 7#1)", "description": ""}}, "parts": [{"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "<p>Evaluate the following expression:</p>\n<p style=\"padding-left: 60px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^T\\) =&nbsp;[[0]]</p>", "gaps": [{"allowFractions": false, "scripts": {}, "markPerCell": false, "correctAnswer": "matrix([\n   [b11,b12],\n   [b21,b22]\n])    ", "allowResize": false, "showCorrectAnswer": true, "tolerance": 0, "extendBaseMarkingAlgorithm": true, "type": "matrix", "useCustomName": false, "marks": "5", "correctAnswerFractions": false, "showFeedbackIcon": true, "numColumns": "2", "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "numRows": "2", "customMarkingAlgorithm": "", "customName": ""}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "sortAnswers": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst"}], "advice": "<p style=\"padding-left: 30px;\">\\(A=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">Remember multiplication of matrices is carried out by multiplying the rows of the first matrix by the columns of the second matrix.</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}=\\begin{pmatrix}\\var{a11}*\\var{a11}+\\var{a12}*\\var{a21}&amp;\\var{a11}*\\var{a12}+\\var{a12}*\\var{a22}\\\\ \\var{a21}*\\var{a11}+\\var{a22}*\\var{a21}&amp;\\var{a21}*\\var{a12}+\\var{a22}*\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{k1}A=\\begin{pmatrix} \\var{k1}*\\var{a11}&amp; \\var{k1}*\\var{a12}\\\\ \\var{k1}*\\var{a21}&amp;\\var{k1}*\\var{a22}\\end{pmatrix}=\\begin{pmatrix} \\simplify{{k1}*{a11}}&amp; \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&amp;\\simplify{{k1}*{a22}}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\left(\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\simplify{{k1}*{a11}}&amp; \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&amp;\\simplify{{k1}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\var{k2}&amp;0\\\\0&amp;\\var{k2}\\end{pmatrix}\\right)^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&amp;\\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}\\\\ \\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}\\)</p>", "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>This question tests students knowledge of basic matrix arithmetic.</p>"}, "variable_groups": [], "type": "question"}, {"name": "John's copy of Matrix multiplication", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["ba21", "a21", "a22", "a23", "b23", "b22", "b21", "ba22", "w3", "w2", "w1", "ba11", "a11", "ba13", "p1", "q1", "q", "q3", "q2", "ab23", "ab22", "ab21", "s3", "s2", "s1", "a12", "s7", "s6", "s5", "s4", "b12", "b13", "b11", "c13", "c12", "c11", "a33", "a32", "a31", "ba32", "ba33", "a13", "ba31", "s10", "s9", "b32", "b31", "v1", "v2", "v3", "mae", "mad", "mag", "mah", "ba12", "b33", "p2", "p3", "ba23", "ab11", "ab31", "ab32", "ab33", "ab12", "m", "s8", "n", "mac", "p", "s", "r", "u", "w", "v", "y", "x", "z", "ab13"], "showQuestionGroupNames": false, "metadata": {"notes": "\n                                                            \t\t                            \t\t<p><strong>5/07/2012:</strong></p>\n                                                            \t\t                            \t\t<p>Added tags.</p>\n                                                            \t\t                            \t\t<p>Question appears to be working correctly.</p>\n                                                            \t\t                            \t\t<p>&nbsp;</p>\n                                                            \t\t                            \n                                                            \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Elementary Exercises in multiplying matrices.&nbsp;</p>"}, "parts": [{"prompt": "\n      \n      \n      <p>Let</p>\n      \n      \n      \n      <p>\\[A = \\left(\\begin{array}{rrr} \\var{a11} &#38; \\var{a12} &#38; \\var{a13}\\\\ \\var{a21} &#38; \\var{a22} &#38; \\var{a23}\\\\ \\var{a31} &#38; \\var{a32} &#38; \\var{a33}\\\\\n      \n      \\end{array}\\right),\\;\\;\\;\\;\n      \n      B=  \\left(\\begin{array}{rrr} \\var{b11} &#38; \\var{b12} &#38; \\var{b13}\\\\ \\var{b21} &#38; \\var{b22} &#38; \\var{b23}\\\\ \\var{b31} &#38; \\var{b32} &#38; \\var{b33}\\\\\n      \n      \\end{array}\\right),\\;\\;\\;\\;\n      \n      v= \\left(\\begin{array}{r} \\var{v1}\\\\ \\var{v2} \\\\ \\var{v3} \\end{array}\\right),\\;\\;\\;\\;\n      \n      w= \\left(\\begin{array}{r} \\var{w1}\\\\ \\var{w2} \\\\ \\var{w3} \\end{array}\\right)\\]</p>\n      \n      \n      \n      <p>Find the following products:</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\">\\[ Av=\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> [[0]] </td>\n      \n      <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\  \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[1]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[2]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td rowspan=\"3\">\\[ Bw=\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> [[3]] </td>\n      \n      <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\  \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[4]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[5]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\">\\[BA=\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> [[6]] </td>\n      \n      <td> [[7]] </td>\n      \n      <td> [[8]]  </td>\n      \n      <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\  \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[9]] </td>\n      \n      <td> [[10]] </td>\n      \n      <td> [[11]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[12]] </td>\n      \n      <td> [[13]] </td>\n      \n      <td> [[14]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"3\">\\[AB=\\left( \\begin{matrix}  \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n      \n      <td> [[15]] </td>\n      \n      <td> [[16]] </td>\n      \n      <td> [[17]]  </td>\n      \n      <td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\  \\end{matrix} \\right.\\] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[18]] </td>\n      \n      <td> [[19]] </td>\n      \n      <td> [[20]] </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td> [[21]] </td>\n      \n      <td> [[22]] </td>\n      \n      <td> [[23]] </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n        \n      ", "type": "gapfill", "gaps": [{"minValue": "{p1}", "maxValue": "{p1}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{p2}", "maxValue": "{p2}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{p3}", "maxValue": "{p3}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{q1}", "maxValue": "{q1}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{q2}", "maxValue": "{q2}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{q3}", "maxValue": "{q3}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba11}", "maxValue": "{ba11}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba12}", "maxValue": "{ba12}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba13}", "maxValue": "{ba13}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba21}", "maxValue": "{ba21}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba22}", "maxValue": "{ba22}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba23}", "maxValue": "{ba23}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba31}", "maxValue": "{ba31}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba32}", "maxValue": "{ba32}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ba33}", "maxValue": "{ba33}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab11}", "maxValue": "{ab11}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab12}", "maxValue": "{ab12}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab13}", "maxValue": "{ab13}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab21}", "maxValue": "{ab21}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab22}", "maxValue": "{ab22}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab23}", "maxValue": "{ab23}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab31}", "maxValue": "{ab31}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab32}", "maxValue": "{ab32}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}, {"minValue": "{ab33}", "maxValue": "{ab33}", "correctAnswerFraction": false, "allowFractions": false, "marks": 0.5, "showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true}, {"prompt": "<p>Consider the following matrices together with the matrices from the first part of the question.</p>\n        <p>\\[\\begin{eqnarray}&amp;C=&amp; \\var{mac},\\;\\;\\;\\; &amp;D=&amp; \\var{mad},\\;\\;\\; \\;&amp;E= &amp;\\var{mae}\\\\&amp;F=&amp; \\left(\\begin{array}{rr} \\var{w1} &amp; \\var{a12}\\\\ \\var{w2} &amp; \\var{b23} \\\\ \\var{w3} &amp; \\var{w2} \\\\\\var{v1} &amp; \\var{b12}\\\\ 0 &amp; \\var{-w2} \\end{array}\\right),\\;\\;\\;\\;&amp;G=&amp;\\var{mag},\\;\\;\\;\\;&amp;H=&amp;\\var{mah} \\end{eqnarray}\\]</p>\n        <p>Which of the following products of matrices can be calculated?</p>\n        <p>[[0]]</p>\n        <p><strong>Please note that for every correct answer you get 0.5 marks and for every incorrect answer 0.5 is taken away. The minimum mark you can get is 0.</strong></p>", "type": "gapfill", "gaps": [{"layout": {"expression": ""}, "scripts": {}, "showCorrectAnswer": true, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "matrix": "v", "type": "m_n_x", "choices": ["<p>$CD$</p>", "<p>$DC$</p>", "<p>$EF$</p>", "<p>$FE$</p>", "<p>$BC$</p>", "<p>$AE$</p>", "<p>$GH$</p>", "<p>$HE$</p>", "<p>$AG$</p>", "<p>$GB$</p>"], "answers": ["<strong>Can be calculated</strong>", "<strong>Cannot be calculated</strong>"], "marks": 0, "maxAnswers": 0, "shuffleAnswers": false, "shuffleChoices": true}], "marks": 0, "scripts": {}, "showCorrectAnswer": true}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "statement": "<p>Answer the following questions on matrices.</p>\n    <p>&nbsp;</p>", "type": "question", "variable_groups": [], "advice": "", "functions": {}, "variables": {"mad": {"group": "Ungrouped variables", "name": "mad", "definition": "matrix(repeat(repeat(random(-2..9),q),p))", "templateType": "anything", "description": ""}, "b21": {"group": "Ungrouped variables", "name": "b21", "definition": "random(0..2)", "templateType": "anything", "description": ""}, "s6": {"group": "Ungrouped variables", "name": "s6", "definition": "if(r=3,0.5,-0.5)", "templateType": "anything", "description": ""}, "s7": {"group": "Ungrouped variables", "name": "s7", "definition": "if(u=y,0.5,-0.5)", "templateType": "anything", "description": ""}, "s8": {"group": "Ungrouped variables", "name": "s8", "definition": "if(x=r,0.5,-0.5)", "templateType": "anything", "description": ""}, "a12": {"group": "Ungrouped variables", "name": "a12", "definition": "random(-1,1,2)", "templateType": "anything", "description": ""}, "y": {"group": "Ungrouped variables", "name": "y", "definition": "u+random(0,z)", "templateType": "anything", "description": ""}, "ab21": {"group": "Ungrouped variables", "name": "ab21", "definition": "a21*b11+a22*b21+a23*b31", "templateType": "anything", "description": ""}, "q3": {"group": "Ungrouped variables", "name": "q3", "definition": "b31*w1+b32*w2+b33*w3", "templateType": "anything", "description": ""}, "c12": {"group": "Ungrouped variables", "name": "c12", "definition": "random(1..3)", "templateType": "anything", "description": ""}, "mac": {"group": "Ungrouped variables", "name": "mac", "definition": "matrix(repeat(repeat(random(-2..9),n),m))", "templateType": "anything", "description": ""}, "b13": {"group": "Ungrouped variables", "name": "b13", "definition": "random(-2..2)", "templateType": "anything", "description": ""}, "ab13": {"group": "Ungrouped variables", "name": "ab13", "definition": "a11*b13+a12*b23+a13*b33", "templateType": "anything", "description": ""}, "z": {"group": "Ungrouped variables", "name": "z", "definition": "random(-2,-1,1,2)", "templateType": "anything", "description": ""}, "s2": {"group": "Ungrouped variables", "name": "s2", "definition": "if(q=m,0.5,-0.5)", "templateType": "anything", "description": ""}, "s5": {"group": "Ungrouped variables", "name": "s5", "definition": "if(m=3,0.5,-0.5)", "templateType": "anything", "description": ""}, "b32": {"group": "Ungrouped variables", "name": "b32", "definition": "random(-3..3)", "templateType": "anything", "description": ""}, "ba23": {"group": "Ungrouped variables", "name": "ba23", "definition": "b21*a13+b22*a23+b23*a33", "templateType": "anything", "description": ""}, "ba21": {"group": "Ungrouped variables", "name": "ba21", "definition": "b21*a11+b22*a21+b23*a31", "templateType": "anything", "description": ""}, "ab31": {"group": "Ungrouped variables", "name": "ab31", "definition": "a31*b11+a32*b21+a33*b31", "templateType": "anything", "description": ""}, "a13": {"group": "Ungrouped variables", "name": "a13", "definition": "random(1..2)", "templateType": "anything", "description": ""}, "p2": {"group": "Ungrouped variables", "name": "p2", "definition": "a21*v1+a22*v2+a23*v3", "templateType": "anything", "description": ""}, "ab12": {"group": "Ungrouped variables", "name": "ab12", "definition": "a11*b12+a12*b22+a13*b32", "templateType": "anything", "description": ""}, "b11": {"group": "Ungrouped variables", "name": "b11", "definition": "random(0,1)", "templateType": "anything", "description": ""}, "q": {"group": "Ungrouped variables", "name": "q", "definition": "m+random(0,z)", "templateType": "anything", "description": ""}, "c11": {"group": "Ungrouped variables", "name": "c11", "definition": "random(-2..2)", "templateType": "anything", "description": ""}, "w1": {"group": "Ungrouped variables", "name": "w1", "definition": "random(4..6)", "templateType": "anything", "description": ""}, "q1": {"group": "Ungrouped variables", "name": "q1", "definition": "b11*w1+b12*w2+b13*w3", "templateType": "anything", "description": ""}, "mag": {"group": "Ungrouped variables", "name": "mag", "definition": "matrix(repeat(repeat(random(-2..9),u),w))", "templateType": "anything", "description": ""}, "ab32": {"group": "Ungrouped variables", "name": "ab32", "definition": "a31*b12+a32*b22+a33*b32", "templateType": "anything", "description": ""}, "q2": {"group": "Ungrouped variables", "name": "q2", "definition": "b21*w1+b22*w2+b23*w3", "templateType": "anything", "description": ""}, "s3": {"group": "Ungrouped variables", "name": "s3", "definition": "if(s=5,0.5,-0.5)", "templateType": "anything", "description": ""}, "s4": {"group": "Ungrouped variables", "name": "s4", "definition": "if(r=2,0.5,-0.5)", "templateType": "anything", "description": ""}, "p": {"group": "Ungrouped variables", "name": "p", "definition": "n+random(0,z)", "templateType": "anything", "description": ""}, "ba31": {"group": "Ungrouped variables", "name": "ba31", "definition": "b31*a11+b32*a21+b33*a31", "templateType": "anything", "description": ""}, "ab33": {"group": "Ungrouped variables", "name": "ab33", "definition": "a31*b13+a32*b23+a33*b33", "templateType": "anything", "description": ""}, "p1": {"group": "Ungrouped variables", "name": "p1", "definition": "a11*v1+a12*v2+a13*v3", "templateType": "anything", "description": ""}, "v1": {"group": "Ungrouped variables", "name": "v1", "definition": "random(-3..3)", "templateType": "anything", "description": ""}, "ab23": {"group": "Ungrouped variables", "name": "ab23", "definition": "a21*b13+a22*b23+a23*b33", "templateType": "anything", "description": ""}, "a32": {"group": "Ungrouped variables", "name": "a32", "definition": "random(-4..4)", "templateType": "anything", "description": ""}, "m": {"group": "Ungrouped variables", "name": "m", "definition": "random(3..4)", "templateType": "anything", "description": ""}, "v": {"group": "Ungrouped variables", "name": "v", "definition": "[[s1,-s1],[s2,-s2],[s3,-s3],[s4,-s4],[s5,-s5],[s6,-s6],[s7,-s7],[s8,-s8],[s9,-s9],[s10,-s10]]", "templateType": "anything", "description": ""}, "a31": {"group": "Ungrouped variables", "name": "a31", "definition": "random(1..3)", "templateType": "anything", "description": ""}, "ba22": {"group": "Ungrouped variables", "name": "ba22", "definition": "b21*a12+b22*a22+b23*a32", "templateType": "anything", "description": ""}, "s": {"group": "Ungrouped variables", "name": "s", "definition": "random(4..6)", "templateType": "anything", "description": ""}, "a11": {"group": "Ungrouped variables", "name": "a11", "definition": "random(1..4)", "templateType": "anything", "description": ""}, "p3": {"group": "Ungrouped variables", "name": "p3", "definition": "a31*v1+a32*v2+a33*v3", "templateType": "anything", "description": ""}, "a33": {"group": "Ungrouped variables", "name": "a33", "definition": "random(1..4)", "templateType": "anything", "description": ""}, "mah": {"group": "Ungrouped variables", "name": "mah", "definition": "matrix(repeat(repeat(random(-2..9),x),y))", "templateType": "anything", "description": ""}, "b31": {"group": "Ungrouped variables", "name": "b31", "definition": "random(1..3)", "templateType": "anything", "description": ""}, "ab11": {"group": "Ungrouped variables", "name": "ab11", "definition": "a11*b11+a12*b21+a13*b31", "templateType": "anything", "description": ""}, "u": {"group": "Ungrouped variables", "name": "u", "definition": "random(3..4)", "templateType": "anything", "description": ""}, "a23": {"group": "Ungrouped variables", "name": "a23", "definition": "random(-2..2)", "templateType": "anything", "description": ""}, "s9": {"group": "Ungrouped variables", "name": "s9", "definition": "if(w=3,0.5,-0.5)", "templateType": "anything", "description": ""}, "c13": {"group": "Ungrouped variables", "name": "c13", "definition": "random(1..2)", "templateType": "anything", "description": ""}, "a21": {"group": "Ungrouped variables", "name": "a21", "definition": "random(-1,0,1)", "templateType": "anything", "description": ""}, "w2": {"group": "Ungrouped variables", "name": "w2", "definition": "random(-4..4)", "templateType": "anything", "description": ""}, "n": {"group": "Ungrouped variables", "name": "n", "definition": "random(3..6 except m)", "templateType": "anything", "description": ""}, "x": {"group": "Ungrouped variables", "name": "x", "definition": "u+random(0,z)", "templateType": "anything", "description": ""}, "ab22": {"group": "Ungrouped variables", "name": "ab22", "definition": "a21*b12+a22*b22+a23*b32", "templateType": "anything", "description": ""}, "ba13": {"group": "Ungrouped variables", "name": "ba13", "definition": "b11*a13+b12*a23+b13*a33", "templateType": "anything", "description": ""}, "w3": {"group": "Ungrouped variables", "name": "w3", "definition": "random(-4..4)", "templateType": "anything", "description": ""}, "a22": {"group": "Ungrouped variables", "name": "a22", "definition": "random(-4,-3,-2,-1,1,2,3,5)", "templateType": "anything", "description": ""}, "mae": {"group": "Ungrouped variables", "name": "mae", "definition": "matrix(repeat(repeat(random(-2..9),s),r))", "templateType": "anything", "description": ""}, "v2": {"group": "Ungrouped variables", "name": "v2", "definition": "random(1..4)", "templateType": "anything", "description": ""}, "r": {"group": "Ungrouped variables", "name": "r", "definition": "random(1..3)", "templateType": "anything", "description": ""}, "ba33": {"group": "Ungrouped variables", "name": "ba33", "definition": "b31*a13+b32*a23+b33*a33", "templateType": "anything", "description": ""}, "ba11": {"group": "Ungrouped variables", "name": "ba11", "definition": "b11*a11+b12*a21+b13*a31", "templateType": "anything", "description": ""}, "s10": {"group": "Ungrouped variables", "name": "s10", "definition": "if(u=3,0.5,-0.5)", "templateType": "anything", "description": ""}, "b33": {"group": "Ungrouped variables", "name": "b33", "definition": "random(-2..2)", "templateType": "anything", "description": ""}, "ba12": {"group": "Ungrouped variables", "name": "ba12", "definition": "b11*a12+b12*a22+b13*a32", "templateType": "anything", "description": ""}, "ba32": {"group": "Ungrouped variables", "name": "ba32", "definition": "b31*a12+b32*a22+b33*a32", "templateType": "anything", "description": ""}, "b12": {"group": "Ungrouped variables", "name": "b12", "definition": "random(0,1)", "templateType": "anything", "description": ""}, "b23": {"group": "Ungrouped variables", "name": "b23", "definition": "random(1,2)", "templateType": "anything", "description": ""}, "w": {"group": "Ungrouped variables", "name": "w", "definition": "random(3..5)", "templateType": "anything", "description": ""}, "v3": {"group": "Ungrouped variables", "name": "v3", "definition": "random(-5..5)", "templateType": "anything", "description": ""}, "b22": {"group": "Ungrouped variables", "name": "b22", "definition": "random(-4,-3,-2,-1,1,2,3,4)", "templateType": "anything", "description": ""}, "s1": {"group": "Ungrouped variables", "name": "s1", "definition": "if(n=p,0.5,-0.5)", "templateType": "anything", "description": ""}}, "tags": ["checked2015", "linear algebra", "mas104220122013CBA4_2", "MAS1602", "MAS2223", "matrices", "matrix", "matrix manipulation", "matrix multiplication", "multiply matrix", "products of matrices"], "variablesTest": {"condition": "", "maxRuns": 100}, "question_groups": [{"questions": [], "name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0}]}, {"name": "John's copy of Find determinants and inverses of 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Let</p>\n<p>\\begin{align} \\mathbf{A} &amp;= \\var{a}, &amp; \\mathbf{B} &amp;= \\var{b}, &amp; \\mathbf{C} &amp;= \\var{c} \\end{align}</p>", "parts": [{"marks": 0, "scripts": {}, "showCorrectAnswer": true, "prompt": "<p>Calculate the determinants of these matrices.</p>\n<p>$\\mathrm{det}\\left(A\\right) = $&nbsp;[[0]]</p>\n<p>$\\mathrm{det}\\left(B\\right) = $&nbsp;[[1]]</p>\n<p>$\\mathrm{det}\\left(C\\right) = $&nbsp;[[2]]</p>\n<p>$\\mathrm{det}\\left(ABC\\right) = $&nbsp;[[3]]</p>", "gaps": [{"allowFractions": false, "maxValue": "det(a)", "scripts": {}, "showCorrectAnswer": true, "minValue": "det(a)", "correctAnswerFraction": false, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "maxValue": "det(b)", "scripts": {}, "showCorrectAnswer": true, "minValue": "det(b)", "correctAnswerFraction": false, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "maxValue": "det(c)", "scripts": {}, "showCorrectAnswer": true, "minValue": "det(c)", "correctAnswerFraction": false, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "maxValue": "det(a*b*c)", "scripts": {}, "showCorrectAnswer": true, "minValue": "det(a*b*c)", "correctAnswerFraction": false, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}], "type": "gapfill"}, {"marks": 0, "scripts": {}, "showCorrectAnswer": true, "prompt": "<p>Find the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals.</p>\n<p>$\\mathbf{A}^{-1} = $&nbsp;[[0]]</p>", "gaps": [{"allowFractions": true, "scripts": {}, "correctAnswer": "inverse(a)", "allowResize": false, "showCorrectAnswer": true, "tolerance": 0, "marks": "2", "correctAnswerFractions": true, "numColumns": "2", "markPerCell": false, "numRows": "2", "type": "matrix"}], "type": "gapfill"}, {"marks": 0, "scripts": {}, "showCorrectAnswer": true, "prompt": "<p>$\\mathbf{B}^{-1} = $&nbsp;[[0]]</p>", "gaps": [{"allowFractions": true, "scripts": {}, "correctAnswer": "inverse(b)", "allowResize": false, "showCorrectAnswer": true, "tolerance": 0, "marks": "2", "correctAnswerFractions": true, "numColumns": "2", "markPerCell": false, "numRows": "2", "type": "matrix"}], "type": "gapfill"}, {"marks": 0, "scripts": {}, "showCorrectAnswer": true, "prompt": "<p>$\\mathbf{C}^{-1} = $&nbsp;[[0]]</p>", "gaps": [{"allowFractions": true, "scripts": {}, "correctAnswer": "inverse(c)", "allowResize": false, "showCorrectAnswer": true, "tolerance": 0, "marks": "2", "correctAnswerFractions": true, "numColumns": "2", "markPerCell": false, "numRows": "2", "type": "matrix"}], "type": "gapfill"}], "type": "question", "ungrouped_variables": ["a11", "a12", "a21", "a22", "b11", "b12", "b21", "b22", "c11", "c12", "c21", "c22", "tr1", "tr2", "tr3", "tr4"], "variable_groups": [{"variables": ["a", "b", "c"], "name": "Unnamed group"}], "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "advice": "<h4>Determinant of a $2 \\times 2$ matrix</h4>\n<p>The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&amp;b \\\\ c&amp;d \\end{pmatrix}$ is given by</p>\n<p>\\[ \\det\\left(\\mathrm{M}\\right) = ad-bc \\]</p>\n<p>If we have two $n \\times n$ matrices $M$ and $N$, then</p>\n<p>\\[ \\det\\left(\\mathrm{MN}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right) \\]</p>\n<p>And it follows that if we have a third matrix $P$,</p>\n<p>\\[ \\det\\left(\\mathrm{MNP}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right)\\det\\left(\\mathrm{P}\\right) \\]</p>\n<h4>a)</h4>\n<p>Thus for our example we have:</p>\n<p>\\begin{align}<br/>\\det\\left(\\mathrm{A}\\right) &amp;= \\simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\\\<br/>\\det\\left(\\mathrm{B}\\right) &amp;= \\simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\\\<br/>\\det\\left(\\mathrm{C}\\right) &amp;= \\simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}}<br/>\\end{align}</p>\n<p>\\begin{align}<br/>\\det\\left( \\mathrm{ABC} \\right) &amp;= \\det(\\mathrm{A}) \\det(\\mathrm{B}) \\det(\\mathrm{C}) \\\\<br/>&amp;= \\simplify[]{{det(a)}*{det(b)}*{det(c)}} \\\\<br/>&amp;= \\var{det(a*b*c)}<br/>\\end{align}</p>\n<h4>Inverse of a $2 \\times 2$ matrix</h4>\n<p>Suppose $\\mathrm{M} = \\begin{pmatrix} a&amp;b \\\\ c&amp;d \\end{pmatrix}$ is a $2 \\times 2$ matrix and $\\det\\left(\\mathrm{M}\\right) = \\Delta \\neq 0$.</p>\n<p>Then $\\mathrm{M}$ is invertible and</p>\n<p>\\[ \\mathrm{M}^{-1} = \\frac{1}{\\Delta} \\begin{pmatrix} d &amp; -b\\\\ -c&amp; a \\end{pmatrix}=\\begin{pmatrix} \\frac{d}{\\Delta} &amp; -\\frac{b}{\\Delta}\\\\ -\\frac{c}{\\Delta}&amp; \\frac{a}{\\Delta} \\end{pmatrix}\\]</p>\n<p>Applying this to these examples we obtain:</p>\n<h4>b)</h4>\n<p>\\[ \\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \\]</p>\n<h4>c)</h4>\n<p>\\[ \\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \\]</p>\n<h4>d)</h4>\n<p>\\[ \\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \\]</p>", "tags": ["checked2015", "determinant of a matrix", "inverse", "inverse matrix", "MAS1602", "matrices", "matrix", "matrix inverse", "matrix multiplication", "multiplication of matrices", "tested1"], "variables": {"tr4": {"name": "tr4", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": ""}, "b": {"name": "b", "templateType": "anything", "group": "Unnamed group", "definition": "matrix([ [b11,b12], [b21,b22] ])", "description": ""}, "c21": {"name": "c21", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": ""}, "c11": {"name": "c11", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,4)", "description": ""}, "b21": {"name": "b21", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": ""}, "b11": {"name": "b11", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(a11=tr2,tr2+1,tr2)", "description": ""}, "c22": {"name": "c22", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(tr4*c11=c21*c12,tr4+1,tr4)", "description": ""}, "tr2": {"name": "tr2", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": ""}, "a21": {"name": "a21", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": ""}, "b12": {"name": "b12", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": ""}, "a22": {"name": "a22", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(tr1*a11=a21*a12,tr1+1,tr1)", "description": ""}, "tr1": {"name": "tr1", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": ""}, "c": {"name": "c", "templateType": "anything", "group": "Unnamed group", "definition": "matrix([ [c11,c12], [c21,c22] ])", "description": ""}, "a11": {"name": "a11", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": ""}, "c12": {"name": "c12", "templateType": "anything", "group": "Ungrouped variables", "definition": "a12+b12", "description": ""}, "tr3": {"name": "tr3", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": ""}, "a": {"name": "a", "templateType": "anything", "group": "Unnamed group", "definition": "matrix([ [a11,a12],[a21,a22] ])", "description": ""}, "b22": {"name": "b22", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(tr3*b11=b21*b12,tr3+1,tr3)", "description": ""}, "a12": {"name": "a12", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "<p><strong>10/07/2012:</strong></p>\n        <p>Added tags.<strong><br /></strong></p>\n        <p>Question appears to be working correctly.</p>\n        <p>Corrected a typo in the Advice section.<strong><br /></strong></p>\n        <p><strong>24/12/2012:</strong></p>\n        <p>Checked calculations, OK. Added tested1 tag.</p>", "description": "<p>Find the determinant and inverse of three $2 \\times 2$ invertible matrices.</p>"}, "functions": {"inverse": {"language": "jme", "type": "matrix", "parameters": [["m", "matrix"]], "definition": "matrix([\n    [m[1][1], -m[0][1]],\n    [-m[1][0], m[0][0]]\n])/det(m)"}}, "showQuestionGroupNames": false}, {"name": "John's copy of Determinant of 3 x 3 matrices", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}, {"name": "Joseph Clarke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2455/"}], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noleadingminus"]}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Let&nbsp;<br/>\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]<br/>Calculate the determinants of these matrices:</p>\n<p>&nbsp;</p>", "functions": {}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "q1", "c", "b", "r1", "q", "p", "p1", "apb", "lcab", "lcabc", "p2", "q2", "apb1", "p3", "q3", "apb2", "lcabc2"], "parts": [{"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "<p>$det(A) = $ [[0]]</p>", "gaps": [{"allowFractions": false, "maxValue": "det(a)", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "customName": "", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "showFeedbackIcon": true, "useCustomName": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "det(a)", "marks": "3", "unitTests": [], "variableReplacements": [], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}, {"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "<p>$det (B)&nbsp;=$&nbsp;[[0]]</p>", "gaps": [{"allowFractions": false, "maxValue": "det(b)", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "customName": "", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "showFeedbackIcon": true, "useCustomName": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "det(b)", "marks": "3", "unitTests": [], "variableReplacements": [], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}, {"scripts": {}, "customName": "", "showCorrectAnswer": true, "prompt": "<p>$det(C)&nbsp;= $ [[0]]</p>\n<p></p>", "gaps": [{"allowFractions": false, "maxValue": "det(C)", "scripts": {}, "correctAnswerStyle": "plain", "showFractionHint": true, "customName": "", "showCorrectAnswer": true, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "showFeedbackIcon": true, "useCustomName": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "det(C)", "marks": "3", "unitTests": [], "variableReplacements": [], "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false}], "advice": "<p>The determinant of a 3 x 3 matrix</p>\n<p>\\[A = \\begin{pmatrix} a_{11} \\ a_{12} \\ a_{13} \\\\ a_{21} \\ a_{22} \\ a_{23} \\\\ a_{31} \\ a_{32} \\ a_{33}&nbsp; &nbsp;\\end{pmatrix}\\]</p>\n<p>is given by</p>\n<p>\\[det(A)&nbsp; = a_{11}\\left| \\begin{matrix} a_{22} \\ a_{23} \\\\ a_{32} \\ a_{33}\\end{matrix}\\right| -&nbsp;a_{12}\\left| \\begin{matrix} a_{21} \\ a_{23} \\\\ a_{31} \\ a_{33}\\end{matrix}\\right| +&nbsp;a_{13}\\left| \\begin{matrix} a_{21} \\ a_{22} \\\\ a_{31} \\ a_{32}\\end{matrix}\\right| &nbsp;\\]</p>\n<p></p>\n<p>This is one way of finding the determinant of a matrix. We can choose any row or column, provided it corresponds with the sign matrix, to calculate the determinant.</p>\n<p></p>\n<p>\\[\\text{Sign matrix} = \\begin{pmatrix}+ \\ - \\ + \\\\ -\\ + \\ - \\\\ + \\ - \\ + \\end{pmatrix} \\]</p>\n<p></p>\n<p>For further information see Section 4 of the Chapter 10 Notes.&nbsp;</p>\n<p></p>", "tags": [], "variables": {"apb": {"name": "apb", "templateType": "anything", "group": "Ungrouped variables", "definition": "a+b", "description": ""}, "lcabc": {"name": "lcabc", "templateType": "anything", "group": "Ungrouped variables", "definition": "p1*a+q1*b-r1*c", "description": ""}, "apb1": {"name": "apb1", "templateType": "anything", "group": "Ungrouped variables", "definition": "p2*a + q2*b", "description": "<p>apb</p>"}, "lcab": {"name": "lcab", "templateType": "anything", "group": "Ungrouped variables", "definition": "p*a+q*b", "description": ""}, "r1": {"name": "r1", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..6 except [0,1,-1,p1,q1])", "description": ""}, "q3": {"name": "q3", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6 except p)", "description": ""}, "p3": {"name": "p3", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6 except p)", "description": "<p>p2</p>"}, "q2": {"name": "q2", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,1,-1,p])", "description": "<p>q</p>"}, "p": {"name": "p", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": ""}, "b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-5..5 except 0),3),3))", "description": ""}, "p1": {"name": "p1", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6 except p)", "description": ""}, "apb2": {"name": "apb2", "templateType": "anything", "group": "Ungrouped variables", "definition": "p3*a + q3*b", "description": "<p>apb</p>"}, "p2": {"name": "p2", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..10 except p1)", "description": ""}, "q": {"name": "q", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..6 except [0,1,-1,p])", "description": ""}, "c": {"name": "c", "templateType": "anything", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-5..5),3),3))", "description": ""}, "q1": {"name": "q1", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..6 except [0,1,-1,p1,q])", "description": ""}, "lcabc2": {"name": "lcabc2", "templateType": "anything", "group": "Ungrouped variables", "definition": "p1*a+q2*b+q1*c", "description": "<p>cabc</p>"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "definition": "matrix(repeat(repeat(random(-5..5 except 0) ,3),3))", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Three examples of determinant of 2x2 matrices.</p>"}, "variable_groups": [], "type": "question"}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": false, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": "<p>Please do not ignore the question: I expect you to have mastered this material.</p>"}, "preventleave": false, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "<p>This test is intended as a bit of revision of the key parts of first year algebra most of which I will not revise in lectures, in paticular:</p>\n<ol>\n<li>Solving linear equations by Gaussian Elimination</li>\n<li>Basic matrix arithmetic</li>\n<li>Matrix multiplication</li>\n<li>Determinants and inverse of 2x2 matrices, and now determinants behave with products</li>\n<li>Determinants of 3x3 matrices</li>\n</ol>\n<p>We will take a systematic look at determinants in chapter 5, but I expect you to be comfortable in calculating them in the 2x2 and 3x3 cases.&nbsp;</p>", "feedbackmessages": []}, "contributors": [{"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}