$A$ a $3 \\times 3$ matrix. Using row operations on the augmented matrix $\\left(A | I_3\\right)$ reduce to $\\left(I_3 | A^{-1}\\right)$.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Find the inverse of the following matrix:
\\[ \\simplify{matrix:A = {a}} \\]
Form the $3 \\times 6$ augmented matrix $\\mathrm{B}$ by placing $I_3$ to the right of $\\mathrm{A}$ as below:
\n\\[\\mathrm{B} = \\left(\\begin{array}{rrr|ccc} \\var{a[0][0]} & \\var{a[0][1]} & \\var{a[0][2]} &\\var{1}&\\var{0}&\\var{0}\\\\ \\var{a[1][0]} & \\var{a[1][1]} & \\var{a[1][2]}&\\var{0}&\\var{1}&\\var{0}\\\\ \\var{a[2][0]} & \\var{a[2][1]} & \\var{a[2][2]}&\\var{0}&\\var{0}&\\var{1}\\\\ \\end{array}\\right)\\]
\nIn subsequent parts work with this matrix using row operations to introduce the identity matrix on the left hand side, with the inverse of $\\mathrm{A}$ eventually appearing on the right hand side.
\nEnter all numbers as fractions or integers, and not as decimals.
", "advice": "The first entry in the second row is $\\var{a[1][0]}$, and we have a $1$ in the first entry of the first row, so multiply the first row by $\\var{abs(zero_2_1)}$ and {if(zero_2_1<0,'subtract from','add to')} row 2 to get
\n\\[ \\simplify[rowvector,all]{{step0[1]}+{zero_2_1}*{step0[0]} = {step1[1]}} \\]
\nThe first entry in the third row is $\\var{a[2][0]}$, so multiply the first row by $\\var{abs(zero_3_1)}$ and {if(zero_3_1<0,'subtract from','add to')} row 3 to get
\n\\[ \\simplify[rowvector,all]{{step0[2]}+{zero_3_1}*{step0[0]} = {step1[2]}} \\]
\nThe matrix at this stage is
\n\\[ \\var{step1} \\]
\nThe second entry in the third row is $\\var{step1[2][1]}$, and we have a $1$ in the second entry of the second row, so multiply the second row by $\\var{abs(zero_3_2)}$ and {if(zero_3_2<0,'subtract from','add to')} row 3 to get
\n\\[ \\simplify[rowvector,all]{{step1[2]}+{zero_3_2}*{step1[1]} = {step1[2]+zero_3_2*step1[1]}} \\]
\n{if(one_3_3<>1,one_3_3_message,'')}
\nThe matrix at this stage is
\n\\[ \\var{step2} \\]
\nThe third entry in the first row is $\\var{step2[0][2]}$, and we have a $1$ in the third entry of the third row, so multiply the third row by $\\var{abs(zero_1_3)}$ and {if(zero_1_3<0,'subtract from','add to')} row 1 to get
\n\\[ \\simplify[rowvector,all]{{step2[0]}+{zero_1_3}*{step2[2]} = {step2[0]+zero_1_3*step2[2]}} \\]
\nThe third entry in the second row is $\\var{step2[1][2]}$, so multiply the third row by $\\var{abs(zero_2_3)}$ and {if(zero_2_3<0,'subtract from','add to')} row 2 to get
\n\\[ \\simplify[rowvector,all]{{step2[1]}+{zero_2_3}*{step2[2]} = {step2[1]+zero_2_3*step2[2]}} \\]
\nThe matrix at this stage is
\n\\[ \\var{step3} \\]
\nThe second entry in the first row is $\\var{step3[0][1]}$, and we have a $1$ in the second entry of the second row, so multiply the second row by $\\var{abs(zero_1_2)}$ and {if(zero_1_2<0,'subtract from','add to')} row 1 to get
\n\\[ \\simplify[rowvector,all]{{step3[0]}+{zero_1_2}*{step3[1]} = {step3[0]+zero_1_2*step3[1]}} \\]
\nThe matrix at this stage is
\n\\[ \\var{step4} \\]
\nNote that we have a $3 \\times 3$ identity matrix in the left three columns, and $\\mathrm{A}^{-1}$ is written in the three columns on the right. That is,
\n\\[ \\mathrm{A}^{-1} = \\var{inverse} \\]
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\nAdditionally, when you introduce zeros in the second and third rows, you should end up with a $1$ in each of the leading columns, so you don't have to do a division.
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", "templateType": "anything", "can_override": false}, "step4": {"name": "step4", "group": "Steps", "definition": "matrix([\n step3[0]+zero_1_2*step3[1],\n step3[1],\n step3[2]\n])", "description": "Make sure row 1 has a zero in the second column.
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\nIf necessary, multiply the second row by a suitable constant so that the second entry in the second row is $1$.
\n$\\left(\\begin{array}{ccc|ccc} \\var{step0[0][0]} & \\var{step0[0][1]} & \\var{step0[0][2]} & \\var{step0[0][3]} & \\var{step0[0][4]} & \\var{step0[0][5]} \\\\ 0 & 1 & ? & ? & ? & ? \\\\ 0 & ? & ? & ? & ? & ? \\end{array}\\right) = $ [[0]]
", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "step1", "correctAnswerFractions": false, "numRows": "3", "numColumns": "6", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Now, using this matrix, introduce a zero in the second column of the third row.
\nIf necessary, multiply the third row by a suitable constant so that the third entry in the third row is $1$.
\n$\\left(\\begin{array}{ccc|ccc} \\var{step0[0][0]} & \\var{step0[0][1]} & \\var{step0[0][2]} & \\var{step0[0][3]} & \\var{step0[0][4]} & \\var{step0[0][5]} \\\\ 0 & 1 & ? & ? & ? & ? \\\\ 0 & 0 & 1 & ? & ? & ? \\end{array}\\right) = $ [[0]]
", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "step2", "correctAnswerFractions": false, "numRows": "3", "numColumns": "6", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Next, introduce zeros into the third column of the second and first rows, by adding suitable multiples of the third row to those rows.
\n$\\left(\\begin{array}{ccc|ccc} ? & ? & 0 & ? & ? & ? \\\\ 0 & 1 & 0 & ? & ? & ? \\\\ 0 & 0 & 1 & ? & ? & ? \\end{array}\\right) = $ [[0]]
", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "step3", "correctAnswerFractions": false, "numRows": "3", "numColumns": "6", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "One final operation is needed: add a multiple of the second row to the first row to introduce a zero in the second column of the first row.
\n\n$\\left(\\begin{array}{ccc|ccc} 1 & 0 & 0 & ? & ? & ? \\\\ 0 & 1 & 0 & ? & ? & ? \\\\ 0 & 0 & 1 & ? & ? & ? \\end{array}\\right) = $ [[0]]
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\n$b = \\var{b}$.
\n$\\vec{x} =$ [[0]]
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