Quiz to assess matrix addition, subtraction, multiplication and multiplication by scalar, determinants and inverses, solving a system of simultaneous equations.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": "80", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Determinant and inverse of 2x2 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "preamble": {"js": "", "css": ""}, "variable_groups": [], "variablesTest": {"condition": "det<>0", "maxRuns": 100}, "functions": {}, "tags": [], "advice": "", "variables": {"a": {"description": "", "definition": "random(-5..5)", "name": "a", "group": "Ungrouped variables", "templateType": "anything"}, "det": {"description": "", "definition": "a*d-b*c", "name": "det", "group": "Ungrouped variables", "templateType": "anything"}, "c": {"description": "", "definition": "random(-5..5)", "name": "c", "group": "Ungrouped variables", "templateType": "anything"}, "M": {"description": "", "definition": "matrix([a,b],[c,d])", "name": "M", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"description": "", "definition": "random(-5..5)", "name": "b", "group": "Ungrouped variables", "templateType": "anything"}, "d": {"description": "", "definition": "random(-5..5)", "name": "d", "group": "Ungrouped variables", "templateType": "anything"}}, "parts": [{"vsetrange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "variableReplacements": [], "marks": 1, "checkvariablenames": false, "answer": "{det}", "type": "jme", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showpreview": false, "prompt": "What is determinant of $A$?
", "showFeedbackIcon": true, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"scripts": {}, "showCorrectAnswer": true, "tolerance": 0, "allowFractions": false, "correctAnswer": "matrix([d,-b],[-c,a])", "variableReplacements": [], "marks": "4", "numColumns": "2", "allowResize": false, "type": "matrix", "variableReplacementStrategy": "originalfirst", "numRows": "2", "markPerCell": true, "correctAnswerFractions": false, "prompt": "Write down the adjoint of $A$. (Swap top-left and bottom-right entries; change signs of top-right and bottom-left entries.)
", "showFeedbackIcon": true}, {"scripts": {}, "showCorrectAnswer": true, "tolerance": 0, "allowFractions": true, "correctAnswer": "matrix([d/det,-b/det],[-c/det,a/det])", "variableReplacements": [], "marks": "4", "numColumns": "2", "allowResize": false, "type": "matrix", "variableReplacementStrategy": "originalfirst", "numRows": "2", "markPerCell": true, "correctAnswerFractions": true, "prompt": "Hence write down the inverse of $A$. Write the entries as fractions or decimals.
", "showFeedbackIcon": true}], "ungrouped_variables": ["M", "a", "b", "c", "d", "det"], "metadata": {"description": "", "licence": "None specified"}, "statement": "The matrix $A$ is:
\n\\[A=\\var{M}\\]
", "rulesets": {}, "type": "question"}, {"name": "Determinant and inverse of 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "variables": {"m11": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m11", "description": ""}, "a22": {"group": "Minors", "templateType": "anything", "definition": "m11*m33-m13*m31", "name": "a22", "description": ""}, "a23": {"group": "Minors", "templateType": "anything", "definition": "m11*m32-m12*m31", "name": "a23", "description": ""}, "m12": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m12", "description": ""}, "m13": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m13", "description": ""}, "m32": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m32", "description": ""}, "M": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([m11,m12,m13],[m21,m22,m23],[m31,m32,m33])", "name": "M", "description": ""}, "a32": {"group": "Minors", "templateType": "anything", "definition": "m11*m23-m13*m21", "name": "a32", "description": ""}, "m23": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m23", "description": ""}, "a21": {"group": "Minors", "templateType": "anything", "definition": "m12*m33-m13*m32", "name": "a21", "description": ""}, "a13": {"group": "Minors", "templateType": "anything", "definition": "m21*m32-m22*m31", "name": "a13", "description": ""}, "m22": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m22", "description": ""}, "a31": {"group": "Minors", "templateType": "anything", "definition": "m12*m23-m13*m22", "name": "a31", "description": ""}, "a12": {"group": "Minors", "templateType": "anything", "definition": "m21*m33-m23*m31", "name": "a12", "description": ""}, "det": {"group": "Ungrouped variables", "templateType": "anything", "definition": "det(M)", "name": "det", "description": ""}, "m33": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m33", "description": ""}, "a33": {"group": "Minors", "templateType": "anything", "definition": "m11*m22-m12*m21", "name": "a33", "description": ""}, "m31": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m31", "description": ""}, "m21": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m21", "description": ""}, "a11": {"group": "Minors", "templateType": "anything", "definition": "m22*m33-m23*m32", "name": "a11", "description": ""}}, "functions": {}, "tags": [], "parts": [{"showFeedbackIcon": true, "vsetrange": [0, 1], "scripts": {}, "vsetrangepoints": 5, "expectedvariablenames": [], "prompt": "What is the determinant of $A$?
", "type": "jme", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "variableReplacements": [], "marks": "4", "checkvariablenames": false, "checkingtype": "absdiff", "showpreview": false, "answer": "{det}"}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": false, "prompt": "Write down the minor matrix of $A$.
", "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": false, "prompt": "Write down the cofactor matrix of $A$. (Certain entries change signs according to pattern below.)
\n\\[\\begin{array}[ccc]++&-&+\\\\-&+&-\\\\+&-&+\\end{array}\\]
\n", "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,-a12,a13],[-a21,a22,-a23],[a31,-a32,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": false, "prompt": "Write down the adjoint of $A$. (Transpose of the cofactor matrix.)
", "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,-a21,a31],[-a12,a22,-a32],[a13,-a23,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}, {"marks": 0, "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showFeedbackIcon": true, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "{det}", "marks": "1", "type": "numberentry", "showCorrectAnswer": true, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "maxValue": "{det}", "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": true, "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,-a21,a31],[-a12,a22,-a32],[a13,-a23,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}], "prompt": "Hence write down the inverse of $A$.
\n| 1 | \n[[1]] | \n
| [[0]] | \n
The matrix $A$ is:
\n\\[A=\\var{M}\\]
", "advice": "", "ungrouped_variables": ["M", "det"], "metadata": {"licence": "None specified", "description": "Determinant, minors, cofactors, adjoint and inverse.
"}, "variablesTest": {"maxRuns": 100, "condition": "det(M)<>0"}, "type": "question"}, {"name": "Simultaneous equations with 3 unknowns", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "Use matrices to solve the following system of equations:
\n\\[\\begin{eqnarray}\\simplify{{M[0][0]}*a+{M[0][1]}*b+{M[0][2]}*c={N[0][0]}}\\\\\\simplify{{M[1][0]}*a+{M[1][1]}*b+{M[1][2]}*c={N[1][0]}}\\\\\\simplify{{M[2][0]}*a+{M[2][1]}*b+{M[2][2]}*c={N[2][0]}}\\end{eqnarray}\\]
", "advice": "The matrix $M$ is formed of the 9 coefficients of $a$, $b$ and $c$ in the equations:
\n\\[M=\\var{M}\\]
\nStart by calculating the determinant of $M$.
\n\\[\\text{det}(M)=\\var{M[0][0]}\\times(\\var{M[1][1]}\\times\\var{M[2][2]}-\\var{M[1][2]}\\times\\var{M[2][1]})-\\var{M[0][1]}\\times(\\var{M[1][0]}\\times\\var{M[2][2]}-\\var{M[1][2]}\\times\\var{M[2][0]})+\\var{M[0][2]}\\times(\\var{M[1][0]}\\times\\var{M[2][1]}-\\var{M[1][1]}\\times\\var{M[2][0]})=\\var{det}\\]
\nNext construct the cofactor matrix:
\n\\[\\var{transpose(Madj)}\\]
\nTranspose this to get the adjoint matrix:
\n\\[\\var{Madj}\\]
\nAnd divide this by the determinant $\\var{det}$ to get the inverse matrix $M^{-1}$:
\n\\[\\var[fractionNumbers]{1/det}\\var{Madj}\\]
\nTo calculate $a$, $b$ and $c$:
\n\\[\\left(\\begin{array}\\\\a\\\\b\\\\c\\end{array}\\right)=M^{-1}\\var{N}=\\var[fractionNumbers]{1/det}\\var{Madj}\\var{N}\\]
\nMultiply the matrices together:
\n\\[\\var{Madj}\\var{N}=\\var{abc*det}\\]
\nDivide by $\\var{det}$:
\n\\[\\left(\\begin{array}\\\\a\\\\b\\\\c\\end{array}\\right)=\\var[fractionNumbers]{1/det}\\var{abc*det}=\\var{abc}\\]
\nTherefore $a=\\var{abc[0][0]}$, $b=\\var{abc[1][0]}$ and $c=\\var{abc[2][0]}$.
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", "correctAnswer": "M", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"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 determinant of the matrix $M$.
\n$\\det(M)=$ [[0]]
\nCalculate the inverse of the matrix $M$.
\n| $M^{-1}=$ | \n1 | \n[[1]] | \n
| [[0]] | \n
Hence calculate the values of the unknowns:
\n