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#### a)

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##### Determinants.
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Here is the formula for determinants:
\$M = \\begin{pmatrix} a & b \\\\ c&d \\end{pmatrix} \\Rightarrow \\mathrm{det}\\left(M\\right) = ad-bc \$

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$\\mathrm{det}\\left(A\\right) = \\simplify[]{{a11}*{a22}-{a12}*{a21} = {dA}}$
$\\mathrm{det}\\left(B\\right) = \\simplify[]{{b11}*{b22}-{b12}*{b21} = {dB}}$
$\\mathrm{det}\\left(C\\right) = \\simplify[]{{c11}*{c22}-{c12}*{c21} = {dC}}$

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##### Determinant of a product of matrices.
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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)\$

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Thus for our example we have:

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\$\\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*} \$

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##### Inverse of a $2 \\times 2$ matrix
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Suppose $M$ is a $2 \\times 2$ matrix and $\\mathrm{det}\\left(M\\right) = \\Delta \\neq 0$.

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Then $M$ 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}\$

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Applying this to these examples we obtain:

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#### b)

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\$A^{-1} = \\begin{pmatrix} \\simplify[std]{{a22}/{dA}} &\\simplify[std]{{-a12}/{dA}}\\\\\\simplify[std]{{-a21}/{dA}}&\\simplify[std]{{a11}/{dA}}\\end{pmatrix}\$

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#### c)

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\$B^{-1} = \\begin{pmatrix} \\simplify[std]{{b22}/{dB}} &\\simplify[std]{{-b12}/{dB}}\\\\\\simplify[std]{{-b21}/{dB}}&\\simplify[std]{{b11}/{dB}}\\end{pmatrix}\$

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#### d)

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\$C^{-1} = \\begin{pmatrix} \\simplify[std]{{c22}/{dC}} &\\simplify[std]{{-c12}/{dC}}\\\\\\simplify[std]{{-c21}/{dC}}&\\simplify[std]{{c11}/{dC}}\\end{pmatrix}\$

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$A^{-1} =$ [[0]]

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$B^{-1} =$ [[0]]

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$C^{-1} =$ [[0]]

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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}\$

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

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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.

\n \t\t \t\t \n \t\t \n \t\t", "description": "

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

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Find the inverses of three $2 \\times 2$ invertible matrices.

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