Please do not ignore the question: I expect you to have mastered this material.
"}}, "duration": 0, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "percentPass": "80", "name": "Maria's copy of Linear Equations and Matrices revision", "showstudentname": true, "metadata": {"description": "A collection of questions on solving equations and revising manipulation of small matrices for 2nd year students ", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": true, "feedback": {"allowrevealanswer": true, "showactualmark": true, "advicethreshold": 0, "showanswerstate": true, "showtotalmark": true, "intro": "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:
\nWe 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.
", "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "question_groups": [{"name": "Revision test for 2nd Year", "pickQuestions": 1, "pickingStrategy": "all-ordered", "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": "Solve the system of equations using Gauss Elimination
\\[\\begin{eqnarray*} &\\var{a}x&+\\;&\\var{a*b-1}y&+\\;\\var{a^2*b-a-a*b}z&=&\\var{c2}\\\\ &\\var{a*c}x&+\\;&\\var{c*b}y&+\\;z&=&\\var{c1}\\\\ &x&+\\;&\\var{b}y&+\\;\\var{b*a-b}z&=&\\var{c3} \\end{eqnarray*} \\]
Part a) Rearrange the order of the equations and represent this as a system of equations using a matrix.
Part b) Introduce zeros in the first column using the first row.
Part c) Introduce zeros in the second
Part d) Solve for $y$ and $x$ using the second and first rows of the reduced matrix.
Re-arrange the rows so that the third row becomes the first row, the first the second and the second the third.
WHY? Choose one of the following:
[[0]]
Now write down the entries of the matrix you will use for Gaussian Elimination, remember to include the constants as the last column.
\n| \\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n[[1]] | \n[[2]] | \n[[3]] | \n[[4]] | \n\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n
| [[5]] | \n[[6]] | \n[[7]] | \n[[8]] | \n||
| [[9]] | \n[[10]] | \n[[11]] | \n[[12]] | \n
To make sure that there is a 1 or -1 in the first row, first column position because that makes the arithmetic easier.
", "Because you always do this swap.
", "To make sure that there is a 1 or -1 in the first row, first column position because that makes the arithmetic more stable.
", "I don't know.
"], "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 \nNow introduce zeros in the first column below the first entry by adding:
[[0]] times the first row to the second row and
[[1]] times the first row to the third row to get the matrix:
| \\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n$\\var{1}$ | \n \n$\\var{b}$ | \n \n$\\var{b*a-b}$ | \n \n$\\var{c3}$ | \n \n\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n
| $\\var{0}$ | \n \n[[2]] | \n \n[[3]] | \n \n[[4]] | \n \n||
| $\\var{0}$ | \n \n[[5]] | \n \n[[6]] | \n \n[[7]] | \n \n
Next multiply the second row by [[8]] to get a 1 in the second entry in the second row.
\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 \nNote that you should have multiplied the second row by a suitable number to get a $1$ in the second entry in the second row.
In this part we introduce a $0$ in the second column below the second entry in the second column by adding:
[[0]] times the second row to the third row to get the matrix:
| \\[\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n$\\var{1}$ | \n \n$\\var{b}$ | \n \n$\\var{b*a-b}$ | \n \n$\\var{c3}$ | \n \n\\[\\left) \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n
| $\\var{0}$ | \n \n$\\var{1}$ | \n \n[[1]] | \n \n[[2]] | \n \n||
| $\\var{0}$ | \n \n$\\var{0}$ | \n \n[[3]] | \n \n[[4]] | \n \n
From this you should find:
\n \n \n \n$z=\\;\\;$[[5]]
\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 \nFrom the second row of the reduced matrix you find an equation involving only $y$ and $z$ and using your value for $z$ we find:
\n \n \n \n$y=\\;\\;$[[0]]
\n \n \n \nThen using the first row we have the equation :
\\[\\simplify[all]{x+ {b}y+{b*a-b}z={c3}}\\]
Using this you can now find $x$:
\n \n \n \n$x=\\;\\;$[[1]]
\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": "Look at the revealed answers for this question. All the information needed is there.
", "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.
"}, "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": "Given the matrix:
\n\\(A=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\)
\n", "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": "Evaluate the following expression:
\n\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^T\\) = [[0]]
", "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": "\\(A=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\)
\n\\(A^2=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\)
\nRemember multiplication of matrices is carried out by multiplying the rows of the first matrix by the columns of the second matrix.
\n\\(A^2=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}=\\begin{pmatrix}\\var{a11}*\\var{a11}+\\var{a12}*\\var{a21}&\\var{a11}*\\var{a12}+\\var{a12}*\\var{a22}\\\\ \\var{a21}*\\var{a11}+\\var{a22}*\\var{a21}&\\var{a21}*\\var{a12}+\\var{a22}*\\var{a22}\\end{pmatrix}\\)
\n\\(A^2=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}\\)
\n\\(\\var{k1}A=\\begin{pmatrix} \\var{k1}*\\var{a11}& \\var{k1}*\\var{a12}\\\\ \\var{k1}*\\var{a21}&\\var{k1}*\\var{a22}\\end{pmatrix}=\\begin{pmatrix} \\simplify{{k1}*{a11}}& \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&\\simplify{{k1}*{a22}}\\end{pmatrix}\\)
\n\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\left(\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\simplify{{k1}*{a11}}& \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&\\simplify{{k1}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\var{k2}&0\\\\0&\\var{k2}\\end{pmatrix}\\right)^t\\)
\n\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&\\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}&\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}^t\\)
\n\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&\\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}\\\\ \\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}&\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}\\)
", "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "This question tests students knowledge of basic matrix arithmetic.
"}, "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\t5/07/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tQuestion appears to be working correctly.
\n \t\t \t\t\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "
Elementary Exercises in multiplying matrices.
"}, "parts": [{"prompt": "\n \n \nLet
\n \n \n \n\\[A = \\left(\\begin{array}{rrr} \\var{a11} & \\var{a12} & \\var{a13}\\\\ \\var{a21} & \\var{a22} & \\var{a23}\\\\ \\var{a31} & \\var{a32} & \\var{a33}\\\\\n \n \\end{array}\\right),\\;\\;\\;\\;\n \n B= \\left(\\begin{array}{rrr} \\var{b11} & \\var{b12} & \\var{b13}\\\\ \\var{b21} & \\var{b22} & \\var{b23}\\\\ \\var{b31} & \\var{b32} & \\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)\\]
\n \n \n \nFind the following products:
\n \n \n \n| \\[ Av=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n[[0]] | \n \n\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n
| [[1]] | \n \n||
| [[2]] | \n \n||
| \\[ Bw=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n[[3]] | \n \n\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n
| [[4]] | \n \n||
| [[5]] | \n \n
| \\[BA=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n[[6]] | \n \n[[7]] | \n \n[[8]] | \n \n\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n
| [[9]] | \n \n[[10]] | \n \n[[11]] | \n \n||
| [[12]] | \n \n[[13]] | \n \n[[14]] | \n \n
| \\[AB=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n[[15]] | \n \n[[16]] | \n \n[[17]] | \n \n\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\] | \n \n
| [[18]] | \n \n[[19]] | \n \n[[20]] | \n \n||
| [[21]] | \n \n[[22]] | \n \n[[23]] | \n \n
Consider the following matrices together with the matrices from the first part of the question.
\n\\[\\begin{eqnarray}&C=& \\var{mac},\\;\\;\\;\\; &D=& \\var{mad},\\;\\;\\; \\;&E= &\\var{mae}\\\\&F=& \\left(\\begin{array}{rr} \\var{w1} & \\var{a12}\\\\ \\var{w2} & \\var{b23} \\\\ \\var{w3} & \\var{w2} \\\\\\var{v1} & \\var{b12}\\\\ 0 & \\var{-w2} \\end{array}\\right),\\;\\;\\;\\;&G=&\\var{mag},\\;\\;\\;\\;&H=&\\var{mah} \\end{eqnarray}\\]
\nWhich of the following products of matrices can be calculated?
\n[[0]]
\nPlease 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.
", "type": "gapfill", "gaps": [{"layout": {"expression": ""}, "scripts": {}, "showCorrectAnswer": true, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "matrix": "v", "type": "m_n_x", "choices": ["$CD$
", "$DC$
", "$EF$
", "$FE$
", "$BC$
", "$AE$
", "$GH$
", "$HE$
", "$AG$
", "$GB$
"], "answers": ["Can be calculated", "Cannot be calculated"], "marks": 0, "maxAnswers": 0, "shuffleAnswers": false, "shuffleChoices": true}], "marks": 0, "scripts": {}, "showCorrectAnswer": true}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "statement": "Answer the following questions on matrices.
\n", "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": "
Let
\n\\begin{align} \\mathbf{A} &= \\var{a}, & \\mathbf{B} &= \\var{b}, & \\mathbf{C} &= \\var{c} \\end{align}
", "parts": [{"marks": 0, "scripts": {}, "showCorrectAnswer": true, "prompt": "Calculate the determinants of these matrices.
\n$\\mathrm{det}\\left(A\\right) = $ [[0]]
\n$\\mathrm{det}\\left(B\\right) = $ [[1]]
\n$\\mathrm{det}\\left(C\\right) = $ [[2]]
\n$\\mathrm{det}\\left(ABC\\right) = $ [[3]]
", "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": "Find the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals.
\n$\\mathbf{A}^{-1} = $ [[0]]
", "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": "$\\mathbf{B}^{-1} = $ [[0]]
", "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": "$\\mathbf{C}^{-1} = $ [[0]]
", "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": "The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is given by
\n\\[ \\det\\left(\\mathrm{M}\\right) = ad-bc \\]
\nIf we have two $n \\times n$ matrices $M$ and $N$, then
\n\\[ \\det\\left(\\mathrm{MN}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right) \\]
\nAnd it follows that if we have a third matrix $P$,
\n\\[ \\det\\left(\\mathrm{MNP}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right)\\det\\left(\\mathrm{P}\\right) \\]
\nThus for our example we have:
\n\\begin{align}
\\det\\left(\\mathrm{A}\\right) &= \\simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\\\
\\det\\left(\\mathrm{B}\\right) &= \\simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\\\
\\det\\left(\\mathrm{C}\\right) &= \\simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}}
\\end{align}
\\begin{align}
\\det\\left( \\mathrm{ABC} \\right) &= \\det(\\mathrm{A}) \\det(\\mathrm{B}) \\det(\\mathrm{C}) \\\\
&= \\simplify[]{{det(a)}*{det(b)}*{det(c)}} \\\\
&= \\var{det(a*b*c)}
\\end{align}
Suppose $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is a $2 \\times 2$ matrix and $\\det\\left(\\mathrm{M}\\right) = \\Delta \\neq 0$.
\nThen $\\mathrm{M}$ is invertible and
\n\\[ \\mathrm{M}^{-1} = \\frac{1}{\\Delta} \\begin{pmatrix} d & -b\\\\ -c& a \\end{pmatrix}=\\begin{pmatrix} \\frac{d}{\\Delta} & -\\frac{b}{\\Delta}\\\\ -\\frac{c}{\\Delta}& \\frac{a}{\\Delta} \\end{pmatrix}\\]
\nApplying this to these examples we obtain:
\n\\[ \\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \\]
\n\\[ \\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \\]
\n\\[ \\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \\]
", "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": "10/07/2012:
\nAdded tags.
Question appears to be working correctly.
\nCorrected a typo in the Advice section.
24/12/2012:
\nChecked calculations, OK. Added tested1 tag.
", "description": "Find the determinant and inverse of three $2 \\times 2$ invertible matrices.
"}, "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": "Let
\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\\]
Calculate the determinants of these matrices:
", "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": "
$det(A) = $ [[0]]
", "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": "$det (B) =$ [[0]]
", "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": "$det(C) = $ [[0]]
\n", "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": "The determinant of a 3 x 3 matrix
\n\\[A = \\begin{pmatrix} a_{11} \\ a_{12} \\ a_{13} \\\\ a_{21} \\ a_{22} \\ a_{23} \\\\ a_{31} \\ a_{32} \\ a_{33} \\end{pmatrix}\\]
\nis given by
\n\\[det(A) = a_{11}\\left| \\begin{matrix} a_{22} \\ a_{23} \\\\ a_{32} \\ a_{33}\\end{matrix}\\right| - a_{12}\\left| \\begin{matrix} a_{21} \\ a_{23} \\\\ a_{31} \\ a_{33}\\end{matrix}\\right| + a_{13}\\left| \\begin{matrix} a_{21} \\ a_{22} \\\\ a_{31} \\ a_{32}\\end{matrix}\\right| \\]
\n\nThis 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.
\n\n\\[\\text{Sign matrix} = \\begin{pmatrix}+ \\ - \\ + \\\\ -\\ + \\ - \\\\ + \\ - \\ + \\end{pmatrix} \\]
\n\nFor further information see Section 4 of the Chapter 10 Notes.
\n", "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": "apb
"}, "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": "p2
"}, "q2": {"name": "q2", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,1,-1,p])", "description": "q
"}, "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": "apb
"}, "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": "cabc
"}, "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": "Three examples of determinant of 2x2 matrices.
"}, "variable_groups": [], "type": "question"}]}], "type": "exam", "contributors": [{"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}