Calculate the determinants of these matrices.
\n$\\mathrm{det}\\left(A\\right) = $ [[0]]
\n$\\mathrm{det}\\left(B\\right) = $ [[1]]
\n$\\mathrm{det}\\left(C\\right) = $ [[2]]
\n$\\mathrm{det}\\left(ABC\\right) = $ [[3]]
", "gaps": [{"showPrecisionHint": false, "scripts": {}, "minValue": "det(a)", "showCorrectAnswer": true, "maxValue": "det(a)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 0.5}, {"showPrecisionHint": false, "scripts": {}, "minValue": "det(b)", "showCorrectAnswer": true, "maxValue": "det(b)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 0.5}, {"showPrecisionHint": false, "scripts": {}, "minValue": "det(c)", "showCorrectAnswer": true, "maxValue": "det(c)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 0.5}, {"showPrecisionHint": false, "scripts": {}, "minValue": "det(a*b*c)", "showCorrectAnswer": true, "maxValue": "det(a*b*c)", "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 0.5}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0}, {"prompt": "Find the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals.
\n$\\mathbf{A}^{-1} = $ [[0]]
", "gaps": [{"allowResize": false, "scripts": {}, "showCorrectAnswer": true, "correctAnswerFractions": true, "markPerCell": false, "marks": "2", "correctAnswer": "inverse(a)", "allowFractions": true, "type": "matrix", "tolerance": 0, "numColumns": "2", "numRows": "2"}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0}, {"prompt": "$\\mathbf{B}^{-1} = $ [[0]]
", "gaps": [{"allowResize": false, "scripts": {}, "showCorrectAnswer": true, "correctAnswerFractions": true, "markPerCell": false, "marks": "2", "correctAnswer": "inverse(b)", "allowFractions": true, "type": "matrix", "tolerance": 0, "numColumns": "2", "numRows": "2"}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0}, {"prompt": "$\\mathbf{C}^{-1} = $ [[0]]
", "gaps": [{"allowResize": false, "scripts": {}, "showCorrectAnswer": true, "correctAnswerFractions": true, "markPerCell": false, "marks": "2", "correctAnswer": "inverse(c)", "allowFractions": true, "type": "matrix", "tolerance": 0, "numColumns": "2", "numRows": "2"}], "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "marks": 0}], "statement": "Let
\n\\begin{align} \\mathbf{A} &= \\var{a}, & \\mathbf{B} &= \\var{b}, & \\mathbf{C} &= \\var{c} \\end{align}
", "name": "Mark's copy of Find determinants and inverses of 2x2 matrices", "tags": ["checked2015", "determinant of a matrix", "inverse", "inverse matrix", "MAS1602", "matrices", "matrix", "matrix inverse", "matrix multiplication", "multiplication of matrices", "tested1"], "advice": "The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is given by
\n\\[ \\det\\left(\\mathrm{M}\\right) = ad-bc \\]
\nIf we have two $n \\times n$ matrices $M$ and $N$, then
\n\\[ \\det\\left(\\mathrm{MN}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right) \\]
\nAnd it follows that if we have a third matrix $P$,
\n\\[ \\det\\left(\\mathrm{MNP}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right)\\det\\left(\\mathrm{P}\\right) \\]
\nThus for our example we have:
\n\\begin{align}
\\det\\left(\\mathrm{A}\\right) &= \\simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\\\
\\det\\left(\\mathrm{B}\\right) &= \\simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\\\
\\det\\left(\\mathrm{C}\\right) &= \\simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}}
\\end{align}
\\begin{align}
\\det\\left( \\mathrm{ABC} \\right) &= \\det(\\mathrm{A}) \\det(\\mathrm{B}) \\det(\\mathrm{C}) \\\\
&= \\simplify[]{{det(a)}*{det(b)}*{det(c)}} \\\\
&= \\var{det(a*b*c)}
\\end{align}
Suppose $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is a $2 \\times 2$ matrix and $\\det\\left(\\mathrm{M}\\right) = \\Delta \\neq 0$.
\nThen $\\mathrm{M}$ is invertible and
\n\\[ \\mathrm{M}^{-1} = \\frac{1}{\\Delta} \\begin{pmatrix} d & -b\\\\ -c& a \\end{pmatrix}=\\begin{pmatrix} \\frac{d}{\\Delta} & -\\frac{b}{\\Delta}\\\\ -\\frac{c}{\\Delta}& \\frac{a}{\\Delta} \\end{pmatrix}\\]
\nApplying this to these examples we obtain:
\n\\[ \\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \\]
\n\\[ \\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \\]
\n\\[ \\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \\]
", "showQuestionGroupNames": false, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "variable_groups": [{"variables": ["a", "b", "c"], "name": "Unnamed group"}], "type": "question", "metadata": {"notes": "10/07/2012:
\nAdded tags.
Question appears to be working correctly.
\nCorrected a typo in the Advice section.
24/12/2012:
\nChecked calculations, OK. Added tested1 tag.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the determinant and inverse of three $2 \\times 2$ invertible matrices.
"}, "preamble": {"js": "", "css": ""}, "contributors": [{"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}]}]}], "contributors": [{"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}]}