// Numbas version: exam_results_page_options {"name": "Matrices: Determinants (Instructional)", "metadata": {"description": "

2x2 by ad-bc, 3x3 by Laplace expansion, properties of determinants

", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 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": "Matrices: Determinants 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Determinant of 2x2 and notation

Students are asked to form the calculation before giving answer.

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

The determinant of the matrix:

$\\large A= \\left( \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right) $ is denoted by $\\large \\left| \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right| $

and is defined to be the number $ad-bc$. That is:

$\\large \\left| \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right| =ad-bc$

We can use the notation $det(A)$ or $|A|$ or $\\Delta$ to denote the determinant of $A$.

", "advice": "

We are asked to calculate the determinants of some $2 \\times 2$ matrices:

We simply have to carry out this calculation for each one,

$\\large \\left| \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right| =ad-bc$

$A=\\var{A}$

$|A|=\\left|\\begin{array}{ccc}\\var{A[0][0]} & \\var{A[0][1]} \\\\\\var{A[1][0]} & \\var{A[1][1]} \\end{array}\\right| =\\var{Apart1}\\large -\\var{Apart2}=\\var{detA}$

$B=\\var{B}$

$|B|=\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[0][1]} \\\\\\var{B[1][0]} & \\var{B[1][1]} \\end{array}\\right| =\\var{Bpart1}\\large -\\var{Bpart2}=\\var{detB}$

$C=\\var{C}$

$|C|= \\left|\\begin{array}{ccc}\\var{C[0][0]} & \\var{C[0][1]} \\\\\\var{C[1][0]} & \\var{C[1][1]} \\end{array}\\right| =\\var{Cpart1}\\large -\\var{Cpart2}=\\var{detC}$

$D=\\var{D}$

$|D|= \\left|\\begin{array}{ccc}\\var{D[0][0]} & \\var{D[0][1]} \\\\\\var{D[1][0]} & \\var{D[1][1]} \\end{array}\\right| =\\var{Dpart1}\\large -\\var{Dpart2}=\\var{detD}$

", "rulesets": {}, "variables": {"n1": {"name": "n1", "group": "Ungrouped variables", "definition": "2", "description": "", "templateType": "number"}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "n1", "description": "", "templateType": "anything"}, "A": {"name": "A", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(1..9),n1),m1)))", "description": "", "templateType": "anything"}, "detA": {"name": "detA", "group": "Matrix A", "definition": "det(A)", "description": "", "templateType": "anything"}, "Apart1": {"name": "Apart1", "group": "Matrix A", "definition": "(A[0][0])*(A[1][1])", "description": "", "templateType": "anything"}, "Apart2": {"name": "Apart2", "group": "Matrix A", "definition": "(A[0][1])*(A[1][0])", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),n1),m1)))", "description": "", "templateType": "anything"}, "Bpart1": {"name": "Bpart1", "group": "Matrix B", "definition": "(B[0][0])*(B[1][1])", "description": "", "templateType": "anything"}, "Bpart2": {"name": "Bpart2", "group": "Matrix B", "definition": "(B[0][1])*(B[1][0])", "description": "", "templateType": "anything"}, "detB": {"name": "detB", "group": "Matrix B", "definition": "det(B)", "description": "", "templateType": "anything"}, "C": {"name": "C", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything"}, "detC": {"name": "detC", "group": "Matrix C", "definition": "det(C)", "description": "", "templateType": "anything"}, "Cpart1": {"name": "Cpart1", "group": "Matrix C", "definition": "(C[0][0])*(C[1][1])", "description": "", "templateType": "anything"}, "Cpart2": {"name": "Cpart2", "group": "Matrix C", "definition": "(C[0][1])*(C[1][0])", "description": "", "templateType": "anything"}, "D": {"name": "D", "group": "Matrix ", "definition": "transpose(matrix(repeat(repeat(random(-9..-1),n1),m1)))", "description": "", "templateType": "anything"}, "Dpart1": {"name": "Dpart1", "group": "Matrix ", "definition": "(D[0][0])*(D[1][1])", "description": "", "templateType": "anything"}, "Dpart2": {"name": "Dpart2", "group": "Matrix ", "definition": "(D[0][1])*(D[1][0])", "description": "", "templateType": "anything"}, "detD": {"name": "detD", "group": "Matrix ", "definition": "det(D)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n1", "m1"], "variable_groups": [{"name": "Matrix A", "variables": ["A", "Apart1", "Apart2", "detA"]}, {"name": "Matrix B", "variables": ["B", "Bpart1", "Bpart2", "detB"]}, {"name": "Matrix C", "variables": ["C", "Cpart1", "Cpart2", "detC"]}, {"name": "Matrix ", "variables": ["D", "Dpart1", "Dpart2", "detD"]}], "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": "

Calculate the determinants of the following matrices:

$A=\\var{A}$

$|A|=\\left|\\begin{array}{ccc}\\var{A[0][0]} & \\var{A[0][1]} \\\\\\var{A[1][0]} & \\var{A[1][1]} \\end{array}\\right| =$ [[0]]$\\large -$[[1]]$=$ [[2]]

$B=\\var{B}$

$|B|=\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[0][1]} \\\\\\var{B[1][0]} & \\var{B[1][1]} \\end{array}\\right| =$ [[3]]$\\large -$[[4]]$=$ [[5]]

$C=\\var{C}$

$|C|= \\left|\\begin{array}{ccc}\\var{C[0][0]} & \\var{C[0][1]} \\\\\\var{C[1][0]} & \\var{C[1][1]} \\end{array}\\right| =$ [[6]]$\\large -$[[7]]$=$ [[8]]

$D=\\var{D}$

$|D|= \\left|\\begin{array}{ccc}\\var{D[0][0]} & \\var{D[0][1]} \\\\\\var{D[1][0]} & \\var{D[1][1]} \\end{array}\\right| =$ [[9]]$\\large -$[[10]]$=$ [[11]]

Determinant of 2x2 and notation

Input answer only.

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

The determinant of the matrix:

$\\large A= \\left( \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right) $ is denoted by $\\large \\left| \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right| $

and is defined to be the number $ad-bc$. That is:

$\\large \\left| \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right| =ad-bc$

We can use the notation $det(A)$ or $|A|$ or $\\Delta$ to denote the determinant of $A$.

", "advice": "

We are asked to calculate the determinants of some $2 \\times 2$ matrices:

We simply have to carry out this calculation for each one,

$\\large \\left| \\begin{array}{ccc} a & b \\\\ c & d \\end{array} \\right| =ad-bc$

$A=\\var{A}$

$|A|=\\left|\\begin{array}{ccc}\\var{A[0][0]} & \\var{A[0][1]} \\\\\\var{A[1][0]} & \\var{A[1][1]} \\end{array}\\right| =\\var{Apart1}\\large -\\var{Apart2}=\\var{detA}$

$B=\\var{B}$

$|B|=\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[0][1]} \\\\\\var{B[1][0]} & \\var{B[1][1]} \\end{array}\\right| =\\var{Bpart1}\\large -\\var{Bpart2}=\\var{detB}$

$C=\\var{C}$

$|C|= \\left|\\begin{array}{ccc}\\var{C[0][0]} & \\var{C[0][1]} \\\\\\var{C[1][0]} & \\var{C[1][1]} \\end{array}\\right| =\\var{Cpart1}\\large -\\var{Cpart2}=\\var{detC}$

$D=\\var{D}$

$|D|= \\left|\\begin{array}{ccc}\\var{D[0][0]} & \\var{D[0][1]} \\\\\\var{D[1][0]} & \\var{D[1][1]} \\end{array}\\right| =\\var{Dpart1}\\large -\\var{Dpart2}=\\var{detD}$

Calculate the determinants of the following matrices:

$A=\\var{A}$

$|A|=\\left|\\begin{array}{ccc}\\var{A[0][0]} & \\var{A[0][1]} \\\\\\var{A[1][0]} & \\var{A[1][1]} \\end{array}\\right| =$ [[0]]

$B=\\var{B}$

$|B|=\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[0][1]} \\\\\\var{B[1][0]} & \\var{B[1][1]} \\end{array}\\right| =$ [[1]]

$C=\\var{C}$

$|C|=\\left|\\begin{array}{ccc}\\var{C[0][0]} & \\var{C[0][1]} \\\\\\var{C[1][0]} & \\var{C[1][1]} \\end{array}\\right|=$ [[2]]

$D=\\var{D}$

$|D|= \\left|\\begin{array}{ccc}\\var{D[0][0]} & \\var{D[0][1]} \\\\\\var{D[1][0]} & \\var{D[1][1]} \\end{array}\\right| =$ [[3]]

", "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": "{detA}", "maxValue": "{detA}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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, "minValue": "{detB}", "maxValue": "{detB}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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, "minValue": "{detC}", "maxValue": "{detC}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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, "minValue": "{detD}", "maxValue": "{detD}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Matrices: Determinants 03 (Laplace Expansion)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Determinant of n x m matrix by Laplace Expansion across top row.

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

The determinant of a square matrix can be obtained by multiplying each element of a chosen row by its corresponding cofactor and then adding these products together.

For example, chosing the first row of a $3 \\times 3$ matrix:

\\[ \\Large \\Delta=a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13}\\]

This technique is known as Laplace Expansion.

", "advice": "

We are asked to use Laplace expansion along the $1^{st}$, top row to calculate the determinant:

\\[ \\Large{\\Delta}=\\left|\\begin{array}{ccc}\\var{a11} & \\var{a12} & \\var{a13} \\\\ \\var{a21} & \\var{a22} & \\var{a23} \\\\ \\var{a31} & \\var{a32} & \\var{a33} \\end{array}\\right| \\]

In this case, the elements in the first row are:

$a_{11}=\\var{a11}$ $a_{12}=\\var{a12}$ $a_{13}=\\var{a13}$

Remember, the minor for each element is formed by \"striking out\" the row and column that the element appears in, leaving a smaller matrix. The co-factor for each one is positive or negative following the pattern:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$+$	$-$	$+$
$-$	$+$	$-$
$+$	$-$	$+$

And so the minors with their co-factors are:

$\\LARGE{A_{11}=+}\\left| \\begin{array}{ccc} \\var{CF11[0][0]} & \\var{CF11[0][1]} \\\\ \\var{CF11[1][0]} & \\var{CF11[1][1]}\\end{array}\\right|=\\var{detCF11}$

$\\LARGE{A_{12}=+}\\left| \\begin{array}{ccc} \\var{CF12[0][0]} & \\var{CF12[0][1]} \\\\ \\var{CF12[1][0]} & \\var{CF12[1][1]}\\end{array}\\right|=-\\var{detCF12}$

$\\LARGE{A_{13}=+}\\left| \\begin{array}{ccc} \\var{CF13[0][0]} & \\var{CF13[0][1]} \\\\ \\var{CF13[1][0]} & \\var{CF13[1][1]}\\end{array}\\right|=\\var{detCF13}$

Now multiply each element by its corresponding cofactor/minor and add these product together:

$ \\Large{\\Delta} =\\Large{(}\\var{a11}\\times\\var{detCF11}\\Large{)+(}\\var{a12}\\times-\\var{detCF12}\\Large{)+(}\\var{a13}\\times\\var{detCF13}\\Large{)}=\\var{detA}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"A": {"name": "A", "group": "Matrix A", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "templateType": "anything", "can_override": false}, "a11": {"name": "a11", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a12": {"name": "a12", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a13": {"name": "a13", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a21": {"name": "a21", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a22": {"name": "a22", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a23": {"name": "a23", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a31": {"name": "a31", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a32": {"name": "a32", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a33": {"name": "a33", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "detA": {"name": "detA", "group": "Matrix A", "definition": "det(A)", "description": "", "templateType": "anything", "can_override": false}, "CF11": {"name": "CF11", "group": "Co-Factors", "definition": "matrix([a22,a23],[a32,a33])", "description": "", "templateType": "anything", "can_override": false}, "CF12": {"name": "CF12", "group": "Co-Factors", "definition": "matrix([a21,a23],[a31,a33])", "description": "", "templateType": "anything", "can_override": false}, "CF13": {"name": "CF13", "group": "Co-Factors", "definition": "matrix([a21,a22],[a31,a32])", "description": "", "templateType": "anything", "can_override": false}, "detCF11": {"name": "detCF11", "group": "Co-Factors", "definition": "det(CF11)", "description": "", "templateType": "anything", "can_override": false}, "detCF12": {"name": "detCF12", "group": "Co-Factors", "definition": "det(CF12)", "description": "", "templateType": "anything", "can_override": false}, "detCF13": {"name": "detCF13", "group": "Co-Factors", "definition": "det(CF13)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Matrix A", "variables": ["A", "a11", "a12", "a13", "a21", "a22", "a23", "a31", "a32", "a33", "detA"]}, {"name": "Co-Factors", "variables": ["CF11", "detCF11", "CF12", "detCF12", "CF13", "detCF13"]}], "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": "

Use Laplace Expansion along the $1^{st}$, top row to calculate the determinant:

\\[ \\Large{\\Delta}=\\left|\\begin{array}{ccc}\\var{a11} & \\var{a12} & \\var{a13} \\\\ \\var{a21} & \\var{a22} & \\var{a23} \\\\ \\var{a31} & \\var{a32} & \\var{a33} \\end{array}\\right| \\]

In this case, the elements in the first row are:

$a_{11}=$ [[0]] $a_{12}=$ [[1]] $a_{13}=$ [[2]]

And so the minors with their co-factors are:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\LARGE{A_{11}=+}$	\n $\\Huge{\|}$ \n	[[3]]	[[4]]	\n $\\Huge{\|}$ \n	$=$	[[7]]
[[5]]	[[6]]

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\LARGE{A_{12}=-}$	\n $\\Huge{\|}$ \n	[[8]]	[[9]]	\n $\\Huge{\|}$ \n	$=$	[[12]]
[[10]]	[[11]]

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\LARGE{A_{13}=+}$	\n $\\Huge{\|}$ \n	[[13]]	[[14]]	\n $\\Huge{\|}$ \n	$=$	[[17]]
[[15]]	[[16]]

Now multiply each element by its corresponding cofactor/minor and add these product together:

$ \\Large{\\Delta} =\\Large{(}$[[18]]$\\times$[[19]]$\\Large{)+(}$[[20]]$\\times$[[21]]$\\Large{)+(}$[[22]]$\\times$[[23]]$\\Large{)}=$[[24]]

", "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": "{a11}", "maxValue": "{a11}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a12}", "maxValue": "{a12}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a13}", "maxValue": "{a13}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a22}", "maxValue": "{a22}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a23}", "maxValue": "{a23}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a32}", "maxValue": "{a32}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a33}", "maxValue": "{a33}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{detCF11}", "maxValue": "{detCF11}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a21}", "maxValue": "{a21}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a23}", "maxValue": "{a23}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a31}", "maxValue": "{a31}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a33}", "maxValue": "{a33}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "-{detCF12}", "maxValue": "-{detCF12}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a21}", "maxValue": "{a21}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a22}", "maxValue": "{a22}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a31}", "maxValue": "{a31}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a32}", "maxValue": "{a32}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{detCF13}", "maxValue": "{detCF13}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a11}", "maxValue": "{a11}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{detCF11}", "maxValue": "{detCF11}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a12}", "maxValue": "{a12}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "-{detCF12}", "maxValue": "-{detCF12}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{a13}", "maxValue": "{a13}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "{detCF13}", "maxValue": "{detCF13}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{detA}", "maxValue": "{detA}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Matrices: Determinants 04 (Properties)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Useful properties of determinants that allow simplification.

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

Determinants exhibit some properties that can usefully be thought of as \"Rules\":

Rule 1: If two rows (or two columns) of a determinant are interchanged then the value of the determinant is multiplied by ($−1$).

Rule 2: The determinant of a matrix $A$ and the determinant of its transpose $A^T$ are equal.

Rule 3: If two rows (or two columns) of a matrix $A$ are equal then it has zero determinant.

Rule 4: If the elements of one row (or one column) of a determinant are multiplied by $k$, then the determinant is also multiplied by $k$.

Rule 5: If we add (or subtract) a multiple of one row (or column) to another, the value of the determinant is unchanged.

Rule 6: The determinant of a lower triangular matrix, an upper triangular matrix or a diagonal matrix is the product of the elements on the leading diagonal.

", "advice": "

We are asked to give the determinants for various matrices without explicitly calculating them. We can do this by applying the \"rules\" given:

$\\large \\left| \\begin{array}{ccc} \\var{a22} & \\var{a11} & \\var{a21}\\\\ \\var{a11} & -7 & \\var{a11} \\\\\\var{a22}& \\var{a11} & \\var{a21} \\end{array} \\right| =0$

Row $1$ and Row $3$ are equal so the determinant is $0$ (by Rule 3).

$\\large \\left| \\begin{array}{ccc} \\var{a11} & 0 & 0 & 0\\\\ \\var{a12} & \\var{a11} & 0 & 0 \\\\ \\var{a22}& \\var{a21}& 1 & 0 \\\\ \\var{a12} & 0 & \\var{a21} & \\var{a22} \\end{array} \\right| =\\simplify{{a11}{a11}{a22}}$

Calculating the determinant of matrices larger than $3 \\times 3$ can be a laborious process. However, this matrix is triangular so its determinant is merely the product of the values in the leading diagonal (by Rule 6).

If $\\large \\left| \\begin{array}{ccc} \\var{a11} & \\var{a12} \\\\ \\var{a21}& \\var{a22} \\end{array} \\right| =\\var{detA1}$ then $\\large \\left| \\begin{array}{ccc} \\var{a21} & \\var{a22} \\\\ \\var{a11}& \\var{a12} \\end{array} \\right| =\\var{detA2}$

In this case, the second matrix has the same elements as the first but Row $1$ and Row $2$ have been interchanged. So the determinant of the second will be the same as the determinant of the first but $\\times -1$ (by Rule 1).

If $\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[0][1]} & \\var{B[0][2]}\\\\ \\var{B[1][0]} & \\var{B[1][1]} & \\var{B[1][2]}\\\\ \\var{B[2][0]} & \\var{B[2][1]}& \\var{B[2][2]}\\end{array}\\right|=\\var{detB}$ then $\\left|\\begin{array}{ccc}\\var{B[0][0]} & \\var{B[1][0]}& \\var{B[2][0]}\\\\ \\var{B[0][1]} & \\var{B[1][1]}& \\var{B[2][1]}\\\\ \\var{B[0][2]} & \\var{B[1][2]}& \\var{B[2][2]}\\end{array}\\right|=\\var{detB}$

You should, hopefully, be able to see that the second matrix is the transpose of the first. Hence their determinants are equal (by Rule 2).

", "rulesets": {}, "variables": {"a11": {"name": "a11", "group": "A Matrices", "definition": "random(-9..9)", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "A Matrices", "definition": "random(-9..9 except a11)", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "A Matrices", "definition": "random(-9..9 except a11 except a12)", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "A Matrices", "definition": "random(-9..9 except a11 except a12 except a21)", "description": "", "templateType": "anything"}, "A1": {"name": "A1", "group": "A Matrices", "definition": "matrix([a11,a12],[a21,a22])", "description": "", "templateType": "anything"}, "A2": {"name": "A2", "group": "A Matrices", "definition": "matrix([a21,a22],[a11,a12])", "description": "", "templateType": "anything"}, "detA1": {"name": "detA1", "group": "A Matrices", "definition": "det(A1)", "description": "", "templateType": "anything"}, "detA2": {"name": "detA2", "group": "A Matrices", "definition": "det(A2)", "description": "", "templateType": "anything"}, "n1": {"name": "n1", "group": "B matrices", "definition": "3", "description": "", "templateType": "number"}, "m1": {"name": "m1", "group": "B matrices", "definition": "n1", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "B matrices", "definition": "transpose(matrix(repeat(repeat(random(-9..9),n1),m1)))", "description": "", "templateType": "anything"}, "detB": {"name": "detB", "group": "B matrices", "definition": "det(B)", "description": "", "templateType": "anything"}, "Bt": {"name": "Bt", "group": "B matrices", "definition": "transpose(B)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "A Matrices", "variables": ["a11", "a12", "a21", "a22", "A1", "A2", "detA1", "detA2"]}, {"name": "B matrices", "variables": ["n1", "m1", "B", "detB", "Bt"]}], "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": "

By using the properties of determinants (above), that is without calculating any determinants, answer the following:

$\\large \\left| \\begin{array}{ccc} \\var{a22} & \\var{a11} & \\var{a21}\\\\ \\var{a11} & -7 & \\var{a11} \\\\\\var{a22}& \\var{a11} & \\var{a21} \\end{array} \\right| =$ [[0]]

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{a11}{a11}{a22}", "maxValue": "{a11}{a11}{a22}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{detA2}", "maxValue": "{detA2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{detB}", "maxValue": "{detB}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "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, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "extensions": [], "custom_part_types": [], "resources": []}

The determinant of the matrix:

We are asked to calculate the determinants of some $2 \\times 2$ matrices:

Calculate the determinants of the following matrices:

The determinant of the matrix:

We are asked to calculate the determinants of some $2 \\times 2$ matrices:

Calculate the determinants of the following matrices:

The determinant of a square matrix can be obtained by multiplying each element of a chosen row by its corresponding cofactor and then adding these products together.

We are asked to use Laplace expansion along the $1^{st}$, top row to calculate the determinant:

Use Laplace Expansion along the $1^{st}$, top row to calculate the determinant:

$\\Huge{|}$

$\\Huge{|}$

$\\Huge{|}$

$\\Huge{|}$

$\\Huge{|}$

$\\Huge{|}$

Determinants exhibit some properties that can usefully be thought of as \"Rules\":

We are asked to give the determinants for various matrices without explicitly calculating them. We can do this by applying the \"rules\" given: