Find the cofactor matrix of the following matrices : [ ccc 5 & 8 …

Question

Find the cofactor matrix of the following matrices : $\left[\begin{array}{ccc}5 & 8 & 7 \\ -1 & -2 & 1 \\ -2 & 1 & 1\end{array}\right]$

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Answer

Let $A =\left(\begin{array}{rrr}5 & 8 & 7 \\ -1 & -2 & 1 \\ -2 & 1 & 1\end{array}\right)$
The cofactor of $a_{i j}$ is given by $A_{i j}=(-1)^{i+j} M _{i j}$
Now, $M _{11}=\left|\begin{array}{rr}-2 & 1 \\ 1 & 1\end{array}\right|=-2-1=-3$
$
\begin{aligned}
& \therefore A_{11}=(-1)^{1+1} M_{11}=1(-3)=-3 \\
& M _{12}=\left|\begin{array}{ll}
-1 & 1 \\
-2 & 1
\end{array}\right|=-1-(-2)=1 \\
& \therefore A _{12}=(-1)^{1+2} M _{12}=-1(1)=-1 \\
& M _{13}=\left|\begin{array}{rr}
-1 & -2 \\
-2 & 1
\end{array}\right|=-1-4=-5 \\
& \therefore A _{13}=(-1)^{1+3} M _{13}=1(-5)=-5 \\
& M_{21}=\left|\begin{array}{ll}
8 & 7 \\
1 & 1
\end{array}\right|=8-7=1 \\
& \therefore A _{21}=(-1)^{2+1} M _{21}=-1(1)=-1 \\
& M _{22}=\left|\begin{array}{rr}
5 & 7 \\
-2 & 1
\end{array}\right|=5+14=19 \\
& \therefore A _{22}=(-1)^{2+2} M _{22}=1(19)=19 \\
& M_{23}=\left|\begin{array}{rr}
5 & 8 \\
-2 & 1
\end{array}\right|=5-(-16)=21 \\
& \therefore A_{23}=(-1)^{2+3} M_{23}=-1(21)=-21 \\
& M_{31}=\left|\begin{array}{rr}
8 & 7 \\
-2 & 1
\end{array}\right|=8-(-14)=22 \\
& \therefore A_{31}=(-1)^{3+1} M_{31}=1(22)=22 \\
& M _{32}=\left|\begin{array}{rr}
5 & 7 \\
-1 & 1
\end{array}\right|=5-(-7)=12 \\
&
\end{aligned}
$
$\begin{aligned} & \therefore A _{32}=(-1)^{3+2} M _{32}=-1(12)=-12 \\ & M_{33}=\left|\begin{array}{rr}5 & 8 \\ -1 & -2\end{array}\right|=-10-(-8)=-2 \\ & \therefore A _{33}=(-1)^{3+3} M _{33}=1(-2)=-2 \\ & \therefore \text { cofactor matrix }=\left(\begin{array}{lll}A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33}\end{array}\right) \\ & =\left(\begin{array}{rrr}-3 & -1 & -5 \\ -1 & 19 & -21 \\ 22 & -12 & -2\end{array}\right) \\ & \end{aligned}$

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