// Numbas version: exam_results_page_options
{"name": "Solving system of equations using row operations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "variables": {"s": {"name": "s", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a21}*{x1}+{a22}*{y1}+{a23}*{z1}"}, "a21": {"name": "a21", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "k*a11"}, "r": {"name": "r", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a11}*{x1}+{a12}*{y1}+{a13}*{z1}"}, "k1": {"name": "k1", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(2..6#1)"}, "d14": {"name": "d14", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{c14}-{c13}*{z1}"}, "a31": {"name": "a31", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "k1*a11"}, "a11": {"name": "a11", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1,2,4,5,10)"}, "b24": {"name": "b24", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{s}-{k}*{r}"}, "b32": {"name": "b32", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{k1}"}, "c14": {"name": "c14", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{r}-{a12}*{b24}"}, "a22": {"name": "a22", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "k*a12+1"}, "y1": {"name": "y1", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..10#1)"}, "b34": {"name": "b34", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{t}-{k1}*{r}"}, "k2": {"name": "k2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a13}-{a12}*({a23}-{k}*{a13})"}, "b23": {"name": "b23", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a23}-{k}*{a13}"}, "d24": {"name": "d24", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{b24}-{b23}*{z1}"}, "a13": {"name": "a13", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(0..10#1)"}, "a12": {"name": "a12", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(2..6#1)"}, "k3": {"name": "k3", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a23}-{k}*{a13}"}, "k": {"name": "k", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(2..5#1)"}, "x1": {"name": "x1", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..10#1)"}, "a32": {"name": "a32", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "k1*(a12+1)"}, "z1": {"name": "z1", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..12#1)"}, "t": {"name": "t", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a31}*{x1}+{a32}*{y1}+{a33}*{z1}"}, "c34": {"name": "c34", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{b34}-{b32}*{b24}"}, "a33": {"name": "a33", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(2..15#1)"}, "a23": {"name": "a23", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "definition": "random(1..8#1)"}, "d34": {"name": "d34", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{z1}"}, "b33": {"name": "b33", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a33}-{k1}*{a13}"}, "c33": {"name": "c33", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{b33}-{b32}*{b23}"}, "c13": {"name": "c13", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{a13}-{a12}*{b23}"}}, "metadata": {"description": "<p>This question asks learners to use row operations to find the inverse of a 3x3 matrix.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "name": "Solving system of equations using row operations", "parts": [{"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "numRows": "3", "showCorrectAnswer": true, "type": "matrix", "markPerCell": false, "numColumns": "4", "allowFractions": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "correctAnswer": "matrix([\n    [{a11},{a12},{a13},{r}],\n    [0,1,{b23},{b24}],\n    [0,{b32},{b33},{b34}]\n])", "correctAnswerFractions": false, "tolerance": 0, "scripts": {}, "marks": "2.5"}], "prompt": "<p>First set up the augmented matrix:</p>\n<p>\\(\\left(\\begin{array}{rrr|c} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}&amp;\\var{r}\\\\\\var{a21}&amp;\\var{a22}&amp;\\var{a23}&amp;\\var{s}\\\\\\var{a31}&amp;\\var{a32}&amp;\\var{a33}&amp;\\var{t}\\\\\\end{array}\\right)\\)</p>\n<p>Use appropriate row operations to get zeroes below the diagonal in the first column as shown in the matrix below&nbsp;on the left and hence enter&nbsp;the correct values into the matrix on the right.</p>\n<p>\\(\\left(\\begin{array}{rrr|c} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}&amp;\\var{r}\\\\0&amp;1&amp;?&amp;?\\\\0&amp;?&amp;?&amp;?\\\\\\end{array}\\right)\\) = &nbsp;[[0]]</p>\n<p></p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "numRows": "3", "showCorrectAnswer": true, "type": "matrix", "markPerCell": false, "numColumns": "4", "allowFractions": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowResize": true, "correctAnswer": "matrix([\n    [{a11},0,{c13},{c14}],\n    [0,1,{b23},{b24}],\n    [0,0,{c33},{c34}]\n])", "correctAnswerFractions": false, "tolerance": 0, "scripts": {}, "marks": "3"}], "prompt": "<p>Taking this matrix apply the appropriate row operations to get zeroes above and below the diagonal in column 2 as shown in the matrix below&nbsp;on the left and hence enter&nbsp;the correct values into the matrix on the right.</p>\n<p>\\(\\left(\\begin{array}{rrr|c} \\var{a11}&amp;0&amp;?&amp;?\\\\0&amp;1&amp;?&amp;?\\\\0&amp;0&amp;?&amp;?\\\\\\end{array}\\right)\\) =[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "numRows": "3", "showCorrectAnswer": true, "type": "matrix", "markPerCell": false, "numColumns": "4", "allowFractions": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowResize": true, "correctAnswer": "matrix([\n    [{a11},0,0,{d14}],\n    [0,1,0,{d24}],\n    [0,0,1,{d34}]\n])", "correctAnswerFractions": false, "tolerance": 0, "scripts": {}, "marks": "1.5"}], "prompt": "<p>Next take this matrix and apply the appropriate row operations to get zeroes above the diagonal in column 3 as shown in the matrix below&nbsp;on the left and hence enter&nbsp;the correct values into the matrix on the right.</p>\n<p>\\(\\left(\\begin{array}{rrr|c} \\var{a11}&amp;0&amp;0&amp;?\\\\0&amp;1&amp;0&amp;?\\\\0&amp;0&amp;1&amp;?\\\\\\end{array}\\right)\\) =&nbsp;[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{x1}", "showCorrectAnswer": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{x1}", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "{y1}", "showCorrectAnswer": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{y1}", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}, {"variableReplacements": [], "minValue": "{z1}", "showCorrectAnswer": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{z1}", "allowFractions": false, "scripts": {}, "marks": "0.5", "correctAnswerStyle": "plain"}], "prompt": "<p>Perform&nbsp;one final row operation and enter your&nbsp;solutions:</p>\n<p style=\"padding-left: 60px; text-align: left;\">\\(x=\\)&nbsp;[[0]]</p>\n<p style=\"padding-left: 60px; text-align: left;\">\\(y=\\)&nbsp;[[1]]</p>\n<p style=\"padding-left: 60px; text-align: left;\">\\(z=\\)&nbsp;[[2]]</p>\n<p style=\"padding-left: 60px; text-align: left;\"></p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "rulesets": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "extensions": [], "statement": "<p>Solve the following syastem of equations using row operations.</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a11}x+\\var{a12}y+\\var{a13}z=\\var{r}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a21}x+\\var{a22}y+\\var{a23}z=\\var{s}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a31}x+\\var{a32}y+\\var{a33}z=\\var{t}\\)</p>\n<p></p>\n<p></p>\n<p></p>", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a11", "a12", "a13", "k", "a21", "a22", "a23", "k1", "a31", "a33", "a32", "k2", "k3", "x1", "y1", "b24", "z1", "r", "s", "t", "b23", "b32", "b33", "b34", "c13", "c14", "c33", "c34", "d14", "d24", "d34"], "advice": "<p>The correct row operations in the first iteration are</p>\n<p style=\"padding-left: 30px;\">new row2 = old row2- \\(\\var{k}*\\)row1</p>\n<p style=\"padding-left: 30px;\">new row3 = old row3- \\(\\var{k1}*\\)row1</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>The correct row operations in the second iteration are</p>\n<p style=\"padding-left: 30px;\">new row1 = old row1 - \\(\\var{a12}*\\)row2</p>\n<p style=\"padding-left: 30px;\">new row3 = old row3 - \\(\\simplify{{a32}-{k1}*{a12}}*\\)row2</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>The correct row operations in the third iteration are</p>\n<p style=\"padding-left: 30px;\">new row1 = old row1 - \\(\\simplify{{a13}-{a12}*({a23}-{k}*{a13})}*\\)row3\\(/\\var{c33}\\)</p>\n<p style=\"padding-left: 30px;\">new row2 = old row2 \\(-(\\simplify{{a23}-{k}*{a13}})*\\)row3\\(/\\var{c33}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}