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{"name": "Multiply 2x2, 2x1 and 1x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"B": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([random(1..4),random(-1,1,2)],[random(-1,0,1),random(-4,-3,-2,-1,1,2,3,5)])", "description": "", "name": "B"}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([[random(-5..5),random(-7..7 except 0)]])", "description": "", "name": "w"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([[random(-2..2)],[random(1..9)]])", "description": "", "name": "v"}}, "ungrouped_variables": ["B", "w", "v"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Multiply 2x2, 2x1 and 1x2 matrices", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "w*v", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": 1, "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>$\\mathbf{w} \\; \\mathbf{v} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "w*B", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": 1, "scripts": {}, "type": "matrix", "numColumns": "2", "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>$\\mathbf{w} \\mathbf{B} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "B*v", "markPerCell": false, "allowResize": false, "correctAnswerFractions": false, "numRows": "2", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>$\\mathbf{B} \\mathbf{v} = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Answer the following questions on matrices.</p>\n<p>Let</p>\n<p>\\begin{align} \\mathbf{v} &amp;= \\var{v}, &amp; \\mathbf{B} &amp;= \\var{B}, &amp; \\mathbf{w} &amp;= \\var{w}&nbsp;\\end{align}</p>", "tags": ["checked2015", "MAS1602", "matrices", "matrix", "matrix multiplication", "matrix product", "multiplication of matrices", "multiply matrix", "product of matrices", "tested1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>10/07/2012:</strong></p>\n<p>Added tags.<strong><br /></strong></p>\n<p>Perhaps it would be worthwhile restraining all components of the three matrices to be non zero?</p>\n<p>Question appears to be working correctly.</p>\n<p><strong>24/12/2012:</strong></p>\n<p>Calculation checked, OK. Added tested1 tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Very elementary matrix multiplication.&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>\\begin{align} <br/>\\mathbf{w} \\mathbf{v} &amp;= \\simplify{{w}*{v}}&nbsp;\\\\ <br/>&amp;= \\simplify[]{{w[0][0]}*{v[0][0]}+{w[0][1]}*{v[1][0]}} \\\\ <br/>&amp;= \\var{w*v} <br/>\\end{align}</p>\n<h4>b)</h4>\n<p>Since $\\mathbf{w}$ is a $1 \\times 2$ matrix and has the same number of columns, $2$, as the rows in the $2 \\times 2$ matrix $\\mathbf{B}$ we can multiply and form&nbsp;$\\mathbf{w} \\mathbf{B}$ which is a $1 \\times 2$ row vector:</p>\n<p>\\begin{align}<br/>\\mathbf{wB} &amp;= \\simplify{{w}*{B}} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {w[0][0]}*{B[0][0]} + {w[0][1]}*{B[1][0]} }, &amp; \\simplify[]{ {w[0][0]}*{B[0][1]} + {w[0][1]}*{B[1][1]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{w*B}<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>Since $\\mathbf{B}$ is a $2 \\times 2$ matrix and has the same number of columns, $2$, as the rows in $\\mathbf{v}$ we can form the matrix $\\mathbf{Bv}$ which will be a $2 \\times 1$ column vector.</p>\n<p>We find this vector to be:</p>\n<p>\\begin{align}<br/>\\mathbf{Bv} &amp;= \\simplify{{B}*{v}} \\\\<br/>&amp;= \\begin{pmatrix} \\simplify[]{ {B[0][0]}*{v[0][0]} + {B[0][1]}*{v[1][0]} } \\\\ \\simplify[]{ {B[1][0]}*{v[0][0]} + {B[1][1]}*{v[1][0]} } \\end{pmatrix} \\\\<br/>&amp;= \\var{B*v}<br/>\\end{align}</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}