Find the inverses of the following matrices by the adjoint mathod… - Vidyadip

Question

Find the inverses of the following matrices by the adjoint mathod : $\left[\begin{array}{cc}2 & -2 \\ 4 & 5\end{array}\right]$

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Answer

$
\begin{aligned}
& \text { Let } A=\left[\begin{array}{cc}
2 & -2 \\
4 & 5
\end{array}\right] \\
& \therefore|A|=\left[\begin{array}{cc}
2 & -2 \\
4 & 5
\end{array}\right]=10+8=18 \neq 0 \\
& \therefore A^{-1} \text { exists. } \\
& A_{11}=(-1)^{1+1} M_{11}=(1)(5)=5 \\
& A_{12}=(-1)^{1+2} M_{12}=(-1)(4)=-4 \\
& A_{21}=(-1)^{2+1} M_{21}=(-1)(-2)=2 \\
& A_{22}=(-1)^{2+2} M_{22}=(1)(2)=2
\end{aligned}
$
$\therefore$ The matrix of the co-factors is
$
\begin{aligned}
& {\left[ A _{ ij }\right]_{2 \times 2}=\left[\begin{array}{ll}
A _{11} & A _{12} \\
A _{21} & A _{22}
\end{array}\right]=\left[\begin{array}{cc}
5 & -4 \\
2 & 2
\end{array}\right]} \\
& \text { Now } \operatorname{adj} A=\left[ A _{ ij }\right]_{2 \times 2}^{\top}=\left[\begin{array}{cc}
5 & 2 \\
-4 & 2
\end{array}\right] \\
& \therefore A -1=\frac{1}{| A |}(\operatorname{adj} A ) \\
& =\frac{1}{18}\left[\begin{array}{cc}
5 & 2 \\
-4 & 2
\end{array}\right] .
\end{aligned}
$

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