Find the determinants of three $2 \\times 2$ invertible matrices.
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\\[M = \\begin{pmatrix} a & b \\\\ c&d \\end{pmatrix} \\Rightarrow \\mathrm{det}\\left(M\\right) = ad-bc \\]
$\\mathrm{det}\\left(A\\right) = \\simplify[]{{a11}*{a22}-{a12}*{a21} = {dA}}$
$\\mathrm{det}\\left(B\\right) = \\simplify[]{{b11}*{b22}-{b12}*{b21} = {dB}}$
$\\mathrm{det}\\left(C\\right) = \\simplify[]{{c11}*{c22}-{c12}*{c21} = {dC}}$
If we have two $n \\times n$ matrices $M$ and $N$ then:
\\[\\mathrm{det}\\left(MN\\right) = \\mathrm{det}\\left(M\\right)\\mathrm{det}\\left(N\\right)\\]
And it follows that if we have a third matrix $P$ that:
\\[\\mathrm{det}\\left(MNP\\right) = \\mathrm{det}\\left(M\\right)\\mathrm{det}\\left(N\\right)\\mathrm{det}\\left(P\\right)\\]
Thus for our example we have:
\n\\[\\begin{eqnarray*}\\mathrm{det}\\left(ABC\\right) &=& \\mathrm{det}\\left(A\\right)\\times\\mathrm{det}\\left(B\\right)\\times\\mathrm{det}\\left(C\\right)\\\\ &=& \\var{dA}\\times \\var{dB} \\times \\var{dC}\\\\ &=& \\var{dA*dB*dC} \\end{eqnarray*} \\]
\nSuppose $M$ is a $2 \\times 2$ matrix and $\\mathrm{det}\\left(M\\right) = \\Delta \\neq 0$.
\nThen $M$ is invertible and:
\\[M = \\begin{pmatrix} a & b \\\\ c&d \\end{pmatrix} \\Rightarrow 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}\\]
Applying this to these examples we obtain:
\n\\[A^{-1} = \\begin{pmatrix} \\simplify[std]{{a22}/{dA}} &\\simplify[std]{{-a12}/{dA}}\\\\\\simplify[std]{{-a21}/{dA}}&\\simplify[std]{{a11}/{dA}}\\end{pmatrix}\\]
\n\\[B^{-1} = \\begin{pmatrix} \\simplify[std]{{b22}/{dB}} &\\simplify[std]{{-b12}/{dB}}\\\\\\simplify[std]{{-b21}/{dB}}&\\simplify[std]{{b11}/{dB}}\\end{pmatrix}\\]
\n\\[C^{-1} = \\begin{pmatrix} \\simplify[std]{{c22}/{dC}} &\\simplify[std]{{-c12}/{dC}}\\\\\\simplify[std]{{-c21}/{dC}}&\\simplify[std]{{c11}/{dC}}\\end{pmatrix}\\]
", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "parts": [{"prompt": "Calculate the determinants of these matrices:
\n$\\mathrm{det}\\left(A\\right) = \\;\\;$[[0]]$,\\;\\;\\;\\mathrm{det}\\left(B\\right) = \\;\\;$[[1]]$,\\;\\;\\;\\mathrm{det}\\left(C\\right) = \\;\\;$[[2]]$,\\;\\;\\;\\mathrm{det}\\left(ABC\\right) = \\;\\;$[[3]]
", "marks": 0, "gaps": [{"allowFractions": false, "scripts": {}, "maxValue": "dA", "minValue": "dA", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "dB", "minValue": "dB", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "dC", "minValue": "dC", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "dA*dB*dC", "minValue": "dA*dB*dC", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 0.5, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "Let
\\[A=\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix},\\;\\; B=\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix},\\;\\; C=\\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}\\]
10/07/2012:
\n \t\t \t\t \t\tAdded tags.
Question appears to be working correctly.
\n \t\t \t\t \t\tCorrected a typo in the Advice section.
Find the determinant of three $2 \\times 2$ invertible matrices.
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