// Numbas version: exam_results_page_options {"name": "Find determinants and inverses of 2x2 matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"type": "question", "tags": ["checked2015", "determinant of a matrix", "inverse", "inverse matrix", "MAS1602", "matrices", "matrix", "matrix inverse", "matrix multiplication", "multiplication of matrices", "tested1"], "parts": [{"marks": 0, "scripts": {}, "gaps": [{"allowFractions": false, "marks": 0.5, "maxValue": "det(a)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "showPrecisionHint": false, "minValue": "det(a)", "type": "numberentry"}, {"allowFractions": false, "marks": 0.5, "maxValue": "det(b)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "showPrecisionHint": false, "minValue": "det(b)", "type": "numberentry"}, {"allowFractions": false, "marks": 0.5, "maxValue": "det(c)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "showPrecisionHint": false, "minValue": "det(c)", "type": "numberentry"}, {"allowFractions": false, "marks": 0.5, "maxValue": "det(a*b*c)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "showPrecisionHint": false, "minValue": "det(a*b*c)", "type": "numberentry"}], "showCorrectAnswer": true, "type": "gapfill", "prompt": "

Calculate the determinants of these matrices.

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$\\mathrm{det}\\left(A\\right) =$ [[0]]

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$\\mathrm{det}\\left(B\\right) =$ [[1]]

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$\\mathrm{det}\\left(C\\right) =$ [[2]]

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$\\mathrm{det}\\left(ABC\\right) =$ [[3]]

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Find the inverses of the following matrices. Input all matrix entries as fractions or integers and not as decimals.

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$\\mathbf{A}^{-1} =$ [[0]]

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$\\mathbf{B}^{-1} =$ [[0]]

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$\\mathbf{C}^{-1} =$ [[0]]

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10/07/2012:

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Question appears to be working correctly.

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Corrected a typo in the Advice section.

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24/12/2012:

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Checked calculations, OK. Added tested1 tag.

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Find the determinant and inverse of three $2 \\times 2$ invertible matrices.

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#### Determinant of a $2 \\times 2$ matrix

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The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is given by

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\$\\det\\left(\\mathrm{M}\\right) = ad-bc \$

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If we have two $n \\times n$ matrices $M$ and $N$, then

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\$\\det\\left(\\mathrm{MN}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right) \$

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And it follows that if we have a third matrix $P$,

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\$\\det\\left(\\mathrm{MNP}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right)\\det\\left(\\mathrm{P}\\right) \$

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#### a)

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Thus for our example we have:

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\\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}

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\\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}

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#### Inverse of a $2 \\times 2$ matrix

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Suppose $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is a $2 \\times 2$ matrix and $\\det\\left(\\mathrm{M}\\right) = \\Delta \\neq 0$.

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Then $\\mathrm{M}$ is invertible and

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\$\\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}\$

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Applying this to these examples we obtain:

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#### b)

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\$\\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \$

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#### c)

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\$\\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \$

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#### d)

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\$\\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \$

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Let

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\\begin{align} \\mathbf{A} &= \\var{a}, & \\mathbf{B} &= \\var{b}, & \\mathbf{C} &= \\var{c} \\end{align}

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