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Do the following matrix problems.

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Let

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\\[\\mathrm{A} = \\var{a},\\;\\; \\mathrm{B} = \\var{b},\\;\\; \\mathrm{C} = \\var{c}\\]

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Calculate the determinants of these matrices:

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

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

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

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

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

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

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

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

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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}

\n

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

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