Question
Find the cofactor matrix of the following matrices : $\left[\begin{array}{cc}1 & 2 \\ 5 & -8\end{array}\right]$

Answer

Let $A =\left(\begin{array}{rr}1 & 2 \\ 5 & -8\end{array}\right)$
Here, $a_{11}=1, M _{11}=-8$
$
\begin{aligned}
& \therefore A _{11}=(-1)^{1+1} M _{11}=-8 \\
& a_{12}=2, M _{12}=5 \\
& \therefore A _{12}=(-1)^{1+2} M _{12}=-1(5)=-5 \\
& a_{21}=5, M _{21}=2 \\
& \therefore A _{21}=(-1)^{2+1} M _{21}=-1(2)=-2 \\
& a_{22}=-8, M _{22}=1 \\
& \therefore A _{22}=(-1)^{2+2} M _{22}=1 . \\
& \therefore \text { cofactor matrix }=\left[\begin{array}{ll}
A _{11} & A _{12} \\
A _{21} & A _{22}
\end{array}\right] \\
& \quad=\left[\begin{array}{rr}
-8 & -5 \\
-2 . & 1
\end{array}\right]
\end{aligned}
$

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