// Numbas version: exam_results_page_options
{"name": "Simon's copy of Matrix multiplication", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<h4>(a)</h4>\n<p>\\[ \\begin{eqnarray*} Av &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}\\\\&nbsp;\\var{a31}&amp;\\var{a32}&amp;\\var{a33}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{v1}\\\\ \\var{v2}\\\\ \\var{v3}\\\\\\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{v1}+{a12}{v2}+{a13}{v3}}\\\\ \\simplify[]{{a21}{v1}+{a22}{v2}+{a23}{v3}}\\\\ \\simplify[]{{a31}{v1}+{a32}{v2}+{a33}{v3}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{p1}\\\\ \\var{p2}\\\\ \\var{p3}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<p></p>\n<p>\\[ \\begin{eqnarray*} Bw &amp;=&amp; \\begin{pmatrix} \\var{b11}&amp;\\var{b12}&amp;\\var{b13}\\\\ \\var{b21}&amp;\\var{b22}&amp;\\var{b23}\\\\&nbsp;\\var{b31}&amp;\\var{b32}&amp;\\var{b33}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{w1}\\\\ \\var{w2}\\\\ \\var{w3}\\\\\\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{b11}{w1}+{b12}{w2}+{b13}{w3}}\\\\ \\simplify[]{{b21}{w1}+{b22}{w2}+{b23}{w3}}\\\\ \\simplify[]{{b31}{w1}+{b32}{w2}+{b33}{w3}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{q1}\\\\ \\var{q2}\\\\ \\var{q3}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<p></p>\n<p>\\[ \\begin{eqnarray*} AB &amp;=&amp; \\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}\\\\&nbsp;\\var{a31}&amp;\\var{a32}&amp;\\var{a33}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&amp;\\var{b12}&amp;\\var{b13}\\\\ \\var{b21}&amp;\\var{b22}&amp;\\var{b23}\\\\&nbsp;\\var{b31}&amp;\\var{b32}&amp;\\var{b33}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\simplify[]{{a11}{b11}+{a12}{b21}+{a13}{b31}}&amp;\\simplify[]{{a11}{b12}+{a12}{b22}+{a13}{b32}}&amp;\\simplify[]{{a11}{b13}+{a12}{b23}+{a13}{b33}}\\\\ \\simplify[]{{a21}{b11}+{a22}{b21}+{a23}{b31}}&amp;\\simplify[]{{a21}{b12}+{a22}{b22}+{a23}{b32}}&amp;\\simplify[]{{a21}{b13}+{a22}{b23}+{a23}{b33}}\\\\ \\simplify[]{{a31}{b11}+{a32}{b21}+{a33}{b31}}&amp;\\simplify[]{{a31}{b12}+{a32}{b22}+{a33}{b32}}&amp;\\simplify[]{{a31}{b13}+{a32}{b23}+{a33}{b33}}\\\\ \\end{pmatrix}\\\\ &amp;=&amp; \\begin{pmatrix} \\var{ab11}&amp;\\var{ab12}&amp;\\var{ab13}\\\\ \\var{ab21}&amp;\\var{ab22}&amp;\\var{ab23}\\\\&nbsp;\\var{ab31}&amp;\\var{ab32}&amp;\\var{ab33}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p><strong></strong>An $m\\times n$ matrix has $m$ rows and $n$ columns</p>\n<p></p>\n<p>If&nbsp;<strong>A</strong> is an $m \\times n$ matrix and&nbsp;<strong>B</strong> is a $p \\times q$ matrix then we can calculate&nbsp;<strong>AB</strong> only if $n =p$, in 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"variableReplacementStrategy": "originalfirst", "maxValue": "{ab23}", "showCorrectAnswer": true, "type": "numberentry", "unitTests": [], "notationStyles": ["plain", "en", "si-en"]}, {"allowFractions": false, "correctAnswerStyle": "plain", "marks": 0.5, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "minValue": "{ab31}", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "maxValue": "{ab31}", "showCorrectAnswer": true, "type": "numberentry", "unitTests": [], "notationStyles": ["plain", "en", "si-en"]}, {"allowFractions": false, "correctAnswerStyle": "plain", "marks": 0.5, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "minValue": "{ab32}", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "maxValue": "{ab32}", "showCorrectAnswer": true, "type": "numberentry", "unitTests": [], "notationStyles": ["plain", "en", "si-en"]}, {"allowFractions": false, "correctAnswerStyle": "plain", "marks": 0.5, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "minValue": "{ab33}", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "maxValue": "{ab33}", "showCorrectAnswer": true, "type": "numberentry", "unitTests": [], "notationStyles": ["plain", "en", "si-en"]}], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Let</p>\n<p>\\[A = \\left(\\begin{array}{rrr} \\var{a11} &amp; \\var{a12} &amp; \\var{a13}\\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23}\\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33}\\\\ \\end{array}\\right),\\;\\;\\;\\; B= \\left(\\begin{array}{rrr} \\var{b11} &amp; \\var{b12} &amp; \\var{b13}\\\\ \\var{b21} &amp; \\var{b22} &amp; \\var{b23}\\\\ \\var{b31} &amp; \\var{b32} &amp; \\var{b33}\\\\ \\end{array}\\right),\\;\\;\\;\\; v= \\left(\\begin{array}{r} \\var{v1}\\\\ \\var{v2} \\\\ \\var{v3} \\end{array}\\right),\\;\\;\\;\\; w= \\left(\\begin{array}{r} \\var{w1}\\\\ \\var{w2} \\\\ \\var{w3} \\end{array}\\right)\\]</p>\n<p>Find the following products:</p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"3\">\\[ Av=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n<td>[[0]]</td>\n<td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>[[2]]</td>\n</tr>\n<tr>\n<td rowspan=\"3\">\\[ Bw=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n<td>[[3]]</td>\n<td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td>[[4]]</td>\n</tr>\n<tr>\n<td>[[5]]</td>\n</tr>\n</tbody>\n</table>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"3\">\\[AB=\\left( \\begin{matrix} \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n<td>[[8]]</td>\n<td rowspan=\"3\">\\[\\left) \\begin{matrix} \\phantom{.} \\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\phantom{.}\\\\ \\end{matrix} \\right.\\]</td>\n</tr>\n<tr>\n<td>[[9]]</td>\n<td>[[10]]</td>\n<td>[[11]]</td>\n</tr>\n<tr>\n<td>[[12]]</td>\n<td>[[13]]</td>\n<td>[[14]]</td>\n</tr>\n</tbody>\n</table>", "sortAnswers": false, "showCorrectAnswer": true, "type": "gapfill", "unitTests": []}, {"marks": 0, "variableReplacements": [], "gaps": [{"shuffleAnswers": false, "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>"], "maxAnswers": 0, "showFeedbackIcon": true, "showCellAnswerState": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minMarks": 0, "variableReplacementStrategy": "originalfirst", "type": "m_n_x", "showCorrectAnswer": true, "minAnswers": 0, "layout": {"type": "all", "expression": ""}, "maxMarks": 0, "unitTests": [], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "scripts": {}, "matrix": "v", "answers": ["<strong>Can be calculated</strong>", "<strong>Cannot be calculated</strong>"], "warningType": "none", "displayType": "radiogroup"}], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "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>", "sortAnswers": false, "showCorrectAnswer": true, "type": "gapfill", "unitTests": []}], "statement": "<p>Answer the following questions on matrices.</p>\n    <p>&nbsp;</p>", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}