// Numbas version: exam_results_page_options
{"name": "Lois's copy of 1.1 Introduction to Matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": [], "name": "Lois's copy of 1.1 Introduction to Matrices", "tags": [], "advice": "<p>a) &nbsp; &nbsp;&nbsp; Remember that the dimensions are $\\rm \\color{red}{rows} \\times \\color{blue}{columns}$, so you need to count the number or rwos and columns in the matrix and wrtite them in that order.</p>\n<p>b) &nbsp; &nbsp;&nbsp; Remember that the elements are in the form $A_{\\rm \\color{red}{ rows},\\color{blue}{columns}}$ where $A$ is the matrix.</p>\n<p>For example if we are looking for $a_{12}$ we look at the matrix $A= \\begin{pmatrix} \\var{a11} &amp;\\bf( \\underline{\\var{a12}}) \\\\ \\var{a21} &amp; \\var{a22}\\end{pmatrix}$ we want the the element on $\\rm \\color{red}{row ~ 1}$ and $\\rm \\color{blue}{column ~ 2}$ which in this case is $\\bf \\underline{\\var{a12}}$</p>\n<p></p>", "rulesets": {}, "parts": [{"prompt": "<p><strong>Example 1 -- </strong>Give the dimensions of the following matrices:</p>\n<p>$A=\\var{A}$ has dimensions [[0]]$ \\times $[[1]]</p>\n<p>$B=\\var{B}$ is a [[2]]$ \\times$[[3]] matrix</p>\n<p>$C=\\var{C}$ is of dimension [[4]]$ \\times$[[5]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{n1}", "minValue": "{n1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{m1}", "minValue": "{m1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{n2}", "minValue": "{n2}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{m2}", "minValue": "{m2}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{n3}", "minValue": "{n3}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{m3}", "minValue": "{m3}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p><strong>Example 2 -- </strong>Give the values of the following elements of the matrices from example 1 above:</p>\n<p>$a_{\\var{n1}\\var{m1}}=$[[0]]</p>\n<p>$b_{\\var{n2}\\var{m2}}=$[[1]]</p>\n<p>$c_{\\var{n3}\\var{m3}}=$[[2]]</p>\n<p>$c_{\\var{n4}\\var{m4}}=$[[3]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "A[n1-1][m1-1]", "minValue": "A[n1-1][m1-1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "b[n2-1][m2-1]", "minValue": "b[n2-1][m2-1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "c[n3-1][m3-1]", "minValue": "c[n3-1][m3-1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "c[n4-1][m4-1]", "minValue": "c[n4-1][m4-1]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>If a matrix is described as being $p \\times q$ then the order is</p>", "matrix": ["1", 0, 0], "shuffleChoices": false, "marks": 0, "variableReplacements": [], "choices": ["<p>$\\mathrm{rows} \\times \\mathrm{columns}$</p>", "<p>$\\mathrm{columns} \\times \\mathrm{rows}$</p>", "<p>Neither of the above.</p>"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", "", ""], "displayColumns": "1", "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"maxAnswers": 0, "prompt": "<p>If a matrix $A$ is described as having elements $a_{ij}$ then the variables on the left can be described as follows:</p>", "matrix": [[0, "0.5", 0], ["0.5", 0, 0]], "shuffleAnswers": false, "minAnswers": 0, "marks": 0, "variableReplacements": [], "answers": ["Columns", "Rows", "Rows or columns interchangeably"], "choices": ["$i$", "$j$"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "warningType": "none", "showCorrectAnswer": true, "type": "m_n_x", "shuffleChoices": false, "minMarks": 0, "layout": {"type": "all", "expression": ""}}], "extensions": [], "statement": "<p>Matrices are rectangular arrays of numbers arranged in rows and columns. &nbsp;For example</p>\n<p>$\\begin{pmatrix} 1&amp;2\\\\ 4&amp;5\\\\ \\end{pmatrix} , &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\begin{pmatrix} 0&amp;-1\\ &amp;3\\\\2&amp;4&amp;-2 \\end{pmatrix}$</p>\n<p>The rows run across the matrices while the columns run down the matrices. &nbsp;Thus in the first matrix above the numbers $\\begin{pmatrix}1&amp;2\\end{pmatrix}$ are in the first row while the numbers $\\begin{pmatrix}4&amp;5\\end{pmatrix}$ are in the second row and similarly the numbers $\\begin{pmatrix} 1\\\\ 4\\\\ \\end{pmatrix}$&nbsp;are in the first column while the numbers $\\begin{pmatrix} 2\\\\ 5\\\\ \\end{pmatrix}$&nbsp;are in the second column.</p>\n<p>A matrix with $\\bf&nbsp;m$ rows and $\\bf n$<strong>&nbsp;</strong>columns is called a matrix of <em>order</em> $\\bf&nbsp;m\\times n$ or <em>dimension</em> $\\bf m\\times&nbsp;n$&nbsp;(or an $\\bf m\\times&nbsp;n$ matrix for&nbsp;brevity).</p>\n<p>When working with matrices the positions of the numbers in the arrays are as important as the actual values of the numbers. &nbsp;Given a matrix called &nbsp;$A$<strong>&nbsp;</strong>the number in row $i$ and column $j$ is usually denoted $a_{ij}$ it is also sometimes called the $ij^{th}$ element of the matrix $A$.</p>", "variable_groups": [{"variables": ["n1", "m1", "A"], "name": "A"}, {"variables": ["n2", "m2", "B"], "name": "B"}, {"variables": ["n3", "m3", "C"], "name": "C"}, {"variables": ["n4", "m4"], "name": "C2"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"A": {"definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "templateType": "anything", "group": "A", "name": "A", "description": ""}, "C": {"definition": "transpose(matrix(repeat(repeat(random(-9..9),n3),m3)))", "templateType": "anything", "group": "C", "name": "C", "description": ""}, "B": {"definition": "transpose(matrix(repeat(repeat(random(-9..9),n2),m2)))", "templateType": "anything", "group": "B", "name": "B", "description": ""}, "m4": {"definition": "random(1..(m3-1) except m3)", "templateType": "anything", "group": "C2", "name": "m4", "description": ""}, "m1": {"definition": "random(1..2)", "templateType": "anything", "group": "A", "name": "m1", "description": ""}, "m3": {"definition": "random(2..4)", "templateType": "anything", "group": "C", "name": "m3", "description": ""}, "m2": {"definition": "random(2..4)", "templateType": "anything", "group": "B", "name": "m2", "description": ""}, "n1": {"definition": "random(1..2)", "templateType": "anything", "group": "A", "name": "n1", "description": ""}, "n2": {"definition": "random(2..4)", "templateType": "anything", "group": "B", "name": "n2", "description": ""}, "n3": {"definition": "random(2..4)", "templateType": "anything", "group": "C", "name": "n3", "description": ""}, "n4": {"definition": "random(1..(n3-1) except n3)", "templateType": "anything", "group": "C2", "name": "n4", "description": ""}}, "metadata": {"description": "<p>a<sub>ij<sup>&nbsp; notation and definition of the order of a matrix.</sup></sub></p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}]}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}