// Numbas version: finer_feedback_settings {"name": "3x3 Matrices", "metadata": {"description": "

This exam test students understanding of determinants and inverses of 3x3 matrices.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Determinant of 3x3 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mark Patterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5064/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}], "tags": [], "metadata": {"description": "

Exercises on calculating the determinant of 3x3 matrices.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

$3 \\times 3$ Matrix Determinants

\n

Calculate the determinants of the following matrices.

", "advice": "

For simplicity, we will use the expansion of the first row to calculate our determinant

\n

a)

\n

We have:

\n

$$
\\begin{aligned}
\\det{A} &= a_1A_1 + b_1B_1+ c_1C_1 \\\\
&= \\var{maA[0][0]}
\\begin{vmatrix}
\\var{maA[1][1]} & \\var{maA[1][2]} \\\\
\\var{maA[2][1]} & \\var{maA[2][2]} \\\\
\\end{vmatrix} - \\var{maA[0][1]} \\begin{vmatrix}
\\var{maA[1][0]} & \\var{maA[1][2]} \\\\
\\var{maA[2][0]} & \\var{maA[2][2]} \\\\
\\end{vmatrix} + \\var{maA[0][2]} \\begin{vmatrix}
\\var{maA[1][0]} & \\var{maA[1][1]} \\\\
\\var{maA[2][0]} & \\var{maA[2][1]} \\\\
\\end{vmatrix} \\\\
&= \\var{maA[0][0]} \\times \\var{a1a} - \\left(\\var{maA[0][1]} \\times \\var{b1a}\\right) + \\var{maA[0][2]} \\times \\var{c1a} \\\\
&= \\var{deta}
\\end{aligned}
$$

\n

b)

\n

We have:

\n

$$
\\begin{aligned}
\\det{B} &= a_1A_1 + b_1B_1+ c_1C_1 \\\\
&= \\var{maB[0][0]}
\\begin{vmatrix}
\\var{maB[1][1]} & \\var{maB[1][2]} \\\\
\\var{maB[2][1]} & \\var{maB[2][2]} \\\\
\\end{vmatrix} - \\var{maB[0][1]} \\begin{vmatrix}
\\var{maB[1][0]} & \\var{maB[1][2]} \\\\
\\var{maB[2][0]} & \\var{maB[2][2]} \\\\
\\end{vmatrix} + \\var{maB[0][2]} \\begin{vmatrix}
\\var{maB[1][0]} & \\var{maB[1][1]} \\\\
\\var{maB[2][0]} & \\var{maB[2][1]} \\\\
\\end{vmatrix} \\\\
&= \\var{maB[0][0]} \\times \\var{a1b} - \\left(\\var{maB[0][1]} \\times \\var{b1b}\\right) + \\var{maB[0][2]} \\times \\var{c1b} \\\\
&= \\var{detb}
\\end{aligned}
$$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"maA": {"name": "maA", "group": "A", "definition": "matrixrandom(3,3)", "description": "", "templateType": "anything", "can_override": false}, "maB": {"name": "maB", "group": "B", "definition": "matrixrandom(3,3)", "description": "", "templateType": "anything", "can_override": false}, "A1a": {"name": "A1a", "group": "A", "definition": "maA[1][1]*maA[2][2]-maA[2][1]*maA[1][2]", "description": "", "templateType": "anything", "can_override": false}, "B1a": {"name": "B1a", "group": "A", "definition": "maA[1][0]*maA[2][2]-maA[2][0]*maA[1][2]", "description": "", "templateType": "anything", "can_override": false}, "C1a": {"name": "C1a", "group": "A", "definition": "maA[1][0]*maA[2][1]-maA[1][1]*maA[2][0]", "description": "", "templateType": "anything", "can_override": false}, "deta": {"name": "deta", "group": "A", "definition": "det(maa)", "description": "", "templateType": "anything", "can_override": false}, "a1b": {"name": "a1b", "group": "Ungrouped variables", "definition": "maB[1][1]*maB[2][2]-maB[2][1]*maB[1][2]", "description": "", "templateType": "anything", "can_override": false}, "b1b": {"name": "b1b", "group": "Ungrouped variables", "definition": "maB[1][0]*maB[2][2]-maB[2][0]*maB[1][2]", "description": "", "templateType": "anything", "can_override": false}, "c1b": {"name": "c1b", "group": "Ungrouped variables", "definition": "maB[1][0]*maB[2][1]-maB[1][1]*maB[2][0]", "description": "", "templateType": "anything", "can_override": false}, "detb": {"name": "detb", "group": "Ungrouped variables", "definition": "det(maB)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1b", "b1b", "c1b", "detb"], "variable_groups": [{"name": "A", "variables": ["maA", "A1a", "B1a", "C1a", "deta"]}, {"name": "B", "variables": ["maB"]}], "functions": {"matrixrandom": {"parameters": [["rows", "integer"], ["columns", "number"]], "type": "matrix", "language": "jme", "definition": "matrix(repeat(repeat(random(0..9)*(random(1,1,1,-1)),columns),rows))"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate the determinant of $A = \\simplify{{maA}}$

", "minValue": "det(maA)", "maxValue": "det(maA)", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate the determinant of $B = \\simplify{{maB}}$

", "minValue": "det(maB)", "maxValue": "det(maB)", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "3x3 Determinant Problems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}], "tags": [], "metadata": {"description": "

This question tests learner's knowledge of the inverse matrix method for a 3x3 matrix.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

$3 \\times 3$ Determinant Problems

\n

Calculate the two values for $x$ that satisfy the equation $|A|=\\var{k}$

\n

where the matrix $A$ is given by:

\n

$$
A=\\begin{pmatrix} x&\\var{a}&\\var{b}\\\\ \\var{c}&x&\\var{d}\\\\\\var{e1}&\\var{f}&\\var{g} \\end{pmatrix}
$$

", "advice": "

We must calculate the determinant as usual, using the expansion of the first row:

\n

$$
\\begin{aligned}
\\det{A} &= x\\begin{vmatrix}
x &\\var{d} \\\\
\\var{f} & \\var{g} \\\\
\\end{vmatrix} -\\var{a} \\begin{vmatrix}
\\var{c} & \\var{d} \\\\
\\var{e1} & \\var{g} \\\\
\\end{vmatrix} + \\var{b} \\begin{vmatrix}
\\var{c} & x \\\\
\\var{e1} & \\var{f}
\\end{vmatrix} \\\\ 
&= x(x-\\simplify{{d}{f}})-\\var{a}(\\simplify{{c}*{g}}-\\simplify{{e1}*{d}})+\\var{b}(\\simplify{{c}*{f}}-\\var{e1}x) \\\\
&= x^2-\\simplify{{d}*{f}}x-\\simplify{{a}*{c}*{g}}+\\simplify{{a}*{e1}*{d}}+\\simplify{{b}*{c}*{f}}-\\simplify{{b}*{e1}}x \\\\
&= x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}}
\\end{aligned}
$$

\n

Since we know that $\\det A = \\var{k}$:

\n

$$
\\begin{aligned}
x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}} &= \\var{k} \\\\
x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{b}*{e1}*{f}*{d}} &= 0
\\end{aligned}
$$

\n

This can be solved by formula or by finding factors:

\n

$$
(x-\\simplify{{f}*{d}})(x-\\simplify{{b}*{e1}})=0
$$

\n

So we have:

\n

$$
x=\\simplify{{f}*{d}}\\,\\,\\,\\,\\,or\\,\\,\\,\\,\\,x=\\simplify{{b}*{e1}}
$$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(6 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(6 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}-{b}*{e1}*{f}*{d}", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "e1", "f", "g", "k"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "

Enter the smaller of the two values

\n

\\(x=\\) [[0]]

\n

Enter the larger of the two values

\n

\\(x=\\) [[1]]

\n

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{f}*{d}", "maxValue": "{f}*{d}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{b}*{e1}", "maxValue": "{b}*{e1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Finding the Transpose", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Tran_Mat.gif", "/srv/numbas/media/question-resources/Tran_Mat.gif"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}], "tags": [], "metadata": {"description": "Transpose of a matrix", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

The Transpose of a Matrix

\n

For each of the given matrices, give the transpose

\n

You will have to define the dimension of the transposed matrix before you enter your answer.

", "advice": "

The transposition process results in rows becoming columns and columns becoming rows.

\n

It may help to imagine the matrix being \"flipped\" about its diagonal.

\n

\n

Using this technique gives us:

\n

a)

\n

$$
A=\\var{A} \\;\\;\\; A^{T}=\\var{TA}
$$         

\n

b)

\n

$$
B=\\var{B} \\;\\;\\; B^{T}=\\var{TB}
$$

\n

c)

\n

$$
C=\\var{C} \\;\\;\\; C^{T}=\\var{TC}
$$

\n

d)

\n

$$
D=\\var{D} \\;\\;\\; D^{T}=\\var{TD}
$$

\n

e)

\n

$$
E=\\var{EE} \\;\\;\\; E^{T}=\\var{TE}
$$     

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"m1": {"name": "m1", "group": "A", "definition": "random(1..2)", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "A", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "B", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "m2": {"name": "m2", "group": "B", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "B", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n2),m2)))", "description": "", "templateType": "anything", "can_override": false}, "n3": {"name": "n3", "group": "C", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "m3": {"name": "m3", "group": "C", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "C": {"name": "C", "group": "C", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n3),m3)))", "description": "", "templateType": "anything", "can_override": false}, "n4": {"name": "n4", "group": "D", "definition": "random(1..(n3-1) except n3)", "description": "", "templateType": "anything", "can_override": false}, "m4": {"name": "m4", "group": "D", "definition": "random(1..(m3-1) except m3)", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "A", "definition": "random(1..(m3-1) except m3)", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "D", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n4),m4)))", "description": "", "templateType": "anything", "can_override": false}, "n5": {"name": "n5", "group": "E", "definition": "m3", "description": "", "templateType": "anything", "can_override": false}, "m5": {"name": "m5", "group": "E", "definition": "n4", "description": "", "templateType": "anything", "can_override": false}, "EE": {"name": "EE", "group": "E", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n5),m5)))", "description": "", "templateType": "anything", "can_override": false}, "TA": {"name": "TA", "group": "A", "definition": "transpose(A)", "description": "", "templateType": "anything", "can_override": false}, "TB": {"name": "TB", "group": "B", "definition": "transpose(B)", "description": "", "templateType": "anything", "can_override": false}, "TC": {"name": "TC", "group": "C", "definition": "transpose(C)", "description": "", "templateType": "anything", "can_override": false}, "TD": {"name": "TD", "group": "D", "definition": "transpose(D)", "description": "", "templateType": "anything", "can_override": false}, "TE": {"name": "TE", "group": "E", "definition": "transpose(EE)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "A", "variables": ["n1", "m1", "A", "TA"]}, {"name": "B", "variables": ["n2", "m2", "B", "TB"]}, {"name": "C", "variables": ["n3", "m3", "C", "TC"]}, {"name": "D", "variables": ["n4", "m4", "D", "TD"]}, {"name": "E", "variables": ["n5", "m5", "EE", "TE"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Let:

\n

$$
A=\\var{A}
$$               

\n

Find $A^{T}$

\n

$A^{T}=$ [[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": "{TA}", "correctAnswerFractions": false, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "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": "

Let:

\n

$$
B=\\var{B}
$$

\n

Find $B^{T}$

\n

$B^{T}=$ [[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": "{TB}", "correctAnswerFractions": false, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "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": "

Let:

\n

$$
C=\\var{C}
$$

\n

Find $C^{T}$

\n

$C^{T}=$ [[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": "{TC}", "correctAnswerFractions": false, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "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": "

Let:

\n

$$
D=\\var{D}
$$

\n

Find $D^{T}$

\n

$D^{T}=$ [[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": "{TD}", "correctAnswerFractions": false, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "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": "

Let:

\n

$$
E=\\var{EE}
$$               

\n

Find $E^{T}$ 

\n

$E^{T}=$ [[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": "{TE}", "correctAnswerFractions": false, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Inverse of a 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}], "tags": [], "metadata": {"description": "

Cofactors Determinant and inverse of a 3x3 matrix.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Inverse of a $3 \\times 3$ Matrix

\n

Follow the steps in the questions to find the inverse of a $3 \\times 3$ matrix

", "advice": "

a)

\n

For simplicity, we will use the expansion of the first row to find the determinant

\n

$$
\\begin{aligned}
\\det{A} &= a_1A_1 + b_1B_1+ c_1C_1 \\\\
&= \\var{matrixA[0][0]}
\\begin{vmatrix}
\\var{matrixA[1][1]} & \\var{matrixA[1][2]} \\\\
\\var{matrixA[2][1]} & \\var{matrixA[2][2]} \\\\
\\end{vmatrix} - \\var{matrixA[0][1]} \\begin{vmatrix}
\\var{matrixA[1][0]} & \\var{matrixA[1][2]} \\\\
\\var{matrixA[2][0]} & \\var{matrixA[2][2]} \\\\
\\end{vmatrix} + \\var{matrixA[0][2]} \\begin{vmatrix}
\\var{matrixA[1][0]} & \\var{matrixA[1][1]} \\\\
\\var{matrixA[2][0]} & \\var{matrixA[2][1]} \\\\
\\end{vmatrix} \\\\
&= \\var{matrixA[0][0]} \\times \\var{cof11} - \\left(\\var{matrixA[0][1]} \\times \\var{cof12}\\right) + \\var{matrixA[0][2]} \\times \\var{cof13} \\\\
&= \\var{deta}
\\end{aligned}
$$

\n

b)

\n

Given arbitrary matrix

\n

$$
A = \\begin{pmatrix}
a & b & c \\\\
d & e & f \\\\
g & h & j \\\\
\\end{pmatrix}
$$

\n

It's cofactors are given by

\n

$$
\\begin{aligned}
A_{11} &=  +\\begin{vmatrix}
e & f \\\\
h & j \\\\
\\end{vmatrix} \\\\
&= +\\begin{vmatrix}
\\var{a22} & \\var{a23} \\\\
\\var{a32} & \\var{a33} \\\\
\\end{vmatrix} \\\\
&= \\var{cof11}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{12} &=  -\\begin{vmatrix}
d & f \\\\
g & j \\\\
\\end{vmatrix} \\\\
&= -\\begin{vmatrix}
\\var{a21} & \\var{a23} \\\\
\\var{a31} & \\var{a33} \\\\
\\end{vmatrix} \\\\
&= \\var{cof12}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{13} &=  +\\begin{vmatrix}
d & e \\\\
g & h \\\\
\\end{vmatrix} \\\\
&= +\\begin{vmatrix}
\\var{a21} & \\var{a22} \\\\
\\var{a31} & \\var{a32} \\\\
\\end{vmatrix} \\\\
&= \\var{cof13}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{21} &=  -\\begin{vmatrix}
b & c \\\\
h & j \\\\
\\end{vmatrix} \\\\
&= -\\begin{vmatrix}
\\var{a12} & \\var{a13} \\\\
\\var{a32} & \\var{a33} \\\\
\\end{vmatrix} \\\\
&= \\var{cof21}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{22} &=  +\\begin{vmatrix}
a & c \\\\
g & j \\\\
\\end{vmatrix} \\\\
&= +\\begin{vmatrix}
\\var{a11} & \\var{a13} \\\\
\\var{a31} & \\var{a33} \\\\
\\end{vmatrix} \\\\
&= \\var{cof22}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{23} &=  -\\begin{vmatrix}
a & b \\\\
g & h \\\\
\\end{vmatrix} \\\\
&= -\\begin{vmatrix}
\\var{a11} & \\var{a12} \\\\
\\var{a31} & \\var{a32} \\\\
\\end{vmatrix} \\\\
&= \\var{cof23}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{31} &=  +\\begin{vmatrix}
b & c \\\\
e & f \\\\
\\end{vmatrix} \\\\
&= +\\begin{vmatrix}
\\var{a12} & \\var{a13} \\\\
\\var{a22} & \\var{a23} \\\\
\\end{vmatrix} \\\\
&= \\var{cof31}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{32} &=  -\\begin{vmatrix}
a & c \\\\
d & f \\\\
\\end{vmatrix} \\\\
&= -\\begin{vmatrix}
\\var{a11} & \\var{a13} \\\\
\\var{a21} & \\var{a23} \\\\
\\end{vmatrix} \\\\
&= \\var{cof32}
\\end{aligned}
$$

\n

$$
\\begin{aligned}
A_{33} &=  +\\begin{vmatrix}
a & b \\\\
d & e \\\\
\\end{vmatrix} \\\\
&= +\\begin{vmatrix}
\\var{a11} & \\var{a12} \\\\
\\var{a21} & \\var{a22} \\\\
\\end{vmatrix} \\\\
&= \\var{cof33}
\\end{aligned}
$$

\n

c)

\n

Using our answer from the previous question, we simply write the cofactors in the form

\n

$$
\\begin{pmatrix}
A_{11} & A_{12} & A_{13} \\\\
A_{21} & A_{22} & A_{23} \\\\
A_{31} & A_{32} & A_{33} \\\\
\\end{pmatrix}
$$

\n

Giving us our matrix of cofactors

\n

$$
\\begin{pmatrix}
\\var{cof11} & \\var{cof12} & \\var{cof13} \\\\
\\var{cof21} & \\var{cof22} & \\var{cof23} \\\\
\\var{cof31} & \\var{cof32} & \\var{cof33} \\\\
\\end{pmatrix}
$$

\n

d)

\n

The transposition process turns rows into columns and columns into rows

\n

Carrying out this process on our matrix of cofactors gives us the adjugate

\n

$$
\\begin{pmatrix}
\\var{cof11} & \\var{cof21} & \\var{cof31} \\\\
\\var{cof12} & \\var{cof22} & \\var{cof32} \\\\
\\var{cof13} & \\var{cof23} & \\var{cof33} \\\\
\\end{pmatrix}
$$

\n

e)

\n

We can find the inverse of $A$ using our determinant and adjugate, using the formula

\n

$$
A^{-1} = \\frac{1}{\\det A}(adj \\; A)
$$

\n

Therefore, we can calculate $A^{-1}$ by

\n

$$
\\begin{aligned}
A^{-1} &= \\frac{1}{\\var{deta}} \\begin{pmatrix}
\\var{cof11} & \\var{cof21} & \\var{cof31} \\\\
\\var{cof12} & \\var{cof22} & \\var{cof32} \\\\
\\var{cof13} & \\var{cof23} & \\var{cof33} \\\\
\\end{pmatrix} \\\\
&= \\var{inverseA}
\\end{aligned}
$$

\n

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"inverseA": {"name": "inverseA", "group": "Ungrouped variables", "definition": "precround(matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/detA,2)", "description": "", "templateType": "anything", "can_override": false}, "cof22": {"name": "cof22", "group": "cofactors", "definition": "a11*a33-a31*a13", "description": "", "templateType": "anything", "can_override": false}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof11": {"name": "cof11", "group": "cofactors", "definition": "a22*a33-a32*a23", "description": "", "templateType": "anything", "can_override": false}, "a32": {"name": "a32", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof23": {"name": "cof23", "group": "cofactors", "definition": "a12*a31-a11*a32", "description": "

cof23

", "templateType": "anything", "can_override": false}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(0..5 except(a21*a12/a11))", "description": "", "templateType": "anything", "can_override": false}, "cof32": {"name": "cof32", "group": "cofactors", "definition": "a13*a21-a11*a23", "description": "", "templateType": "anything", "can_override": false}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof13": {"name": "cof13", "group": "cofactors", "definition": "a21*a32-a31*a22", "description": "", "templateType": "anything", "can_override": false}, "matrixA": {"name": "matrixA", "group": "Ungrouped variables", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "templateType": "anything", "can_override": false}, "a31": {"name": "a31", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof21": {"name": "cof21", "group": "cofactors", "definition": "a32*a13-a12*a33", "description": "", "templateType": "anything", "can_override": false}, "a13": {"name": "a13", "group": "Ungrouped variables", "definition": "random(-5..10)", "description": "", "templateType": "anything", "can_override": false}, "cof12": {"name": "cof12", "group": "cofactors", "definition": "a23*a31-a21*a33", "description": "", "templateType": "anything", "can_override": false}, "cof33": {"name": "cof33", "group": "cofactors", "definition": "a11*a22-a12*a21", "description": "", "templateType": "anything", "can_override": false}, "cof31": {"name": "cof31", "group": "cofactors", "definition": "a12*a23-a22*a13", "description": "", "templateType": "anything", "can_override": false}, "detA": {"name": "detA", "group": "Ungrouped variables", "definition": "a11*cof11+a12*cof12+a13*cof13", "description": "", "templateType": "anything", "can_override": false}, "a23": {"name": "a23", "group": "Ungrouped variables", "definition": "random(-4..4)", "description": "", "templateType": "anything", "can_override": false}, "a33": {"name": "a33", "group": "Ungrouped variables", "definition": "random(0..20)", "description": "", "templateType": "anything", "can_override": false}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "deta <> 0", "maxRuns": 100}, "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "inverseA", "detA"], "variable_groups": [{"name": "cofactors", "variables": ["cof11", "cof12", "cof13", "cof21", "cof22", "cof23", "cof31", "cof32", "cof33"]}, {"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "

Let:

\n

$$
A=\\var{matrixA}
$$

\n

Find the determinant of $A$

\n

$\\det A =$ [[0]]

", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "det a", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "det(matrixA)", "maxValue": "det(matrixA)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "

Calculate the nine cofactors of $A$

\n

The cofactor $A_{ij}$ denotes the cofactor in row $i$ and column $j$

\n

$A _{11}=$ [[0]]

\n

$A_{12}=$ [[1]]

\n

$A_{13}=$ [[2]]

\n

$A_{21}=$ [[3]]

\n

$A_{22}=$ [[4]]

\n

$A_{23}=$ [[5]]

\n

$A_{31}=$ [[6]]

\n

$A_{32}=$ [[7]]

\n

$A_{33}=$ [[8]]

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof11}", "maxValue": "{cof11}", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof12}", "maxValue": "{cof12}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof13}", "maxValue": "{cof13}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof21}", "maxValue": "{cof21}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof22}", "maxValue": "{cof22}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof23}", "maxValue": "{cof23}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof31}", "maxValue": "{cof31}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof32}", "maxValue": "{cof32}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof33}", "maxValue": "{cof33}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "

Using your answer to part $b)$, state the matrix of cofactors

\n

[[0]]

", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([cof11,cof12,cof12],[cof21,cof22,cof32],[cof31,cof32,cof33])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "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": "

Find the transpose of your matrix from part $c)$, giving us the adjugate 

\n

[[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": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "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": "

Using your answers from all the previous parts, find the inverse of $A$

\n

Elements will be accepted as fractions or correct to 2 decimal places

\n

$A^{-1}=$ [[0]]

", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "inv a", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "cof11", "part": "p1g0", "must_go_first": true}, {"variable": "cof12", "part": "p1g1", "must_go_first": true}, {"variable": "cof13", "part": "p1g2", "must_go_first": true}, {"variable": "cof21", "part": "p1g3", "must_go_first": true}, {"variable": "cof22", "part": "p1g4", "must_go_first": true}, {"variable": "cof23", "part": "p1g5", "must_go_first": true}, {"variable": "cof31", "part": "p1g6", "must_go_first": true}, {"variable": "cof32", "part": "p1g7", "must_go_first": true}, {"variable": "cof33", "part": "p1g8", "must_go_first": true}, {"variable": "detA", "part": "p0g0", "must_go_first": true}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/det(matrixA)", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": "0.005", "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "typeendtoleave": false, "startpassword": "", "allowAttemptDownload": false, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "inreview"}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "type": "exam", "contributors": [{"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/Tran_Mat.gif", "/srv/numbas/media/question-resources/Tran_Mat.gif"]]}