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  4. If A =[4 -5 -2 5 -4 2 2 2 8], then adj. (A) equals:

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Q. If $A =\begin{bmatrix}4 & -5 & -2 \\ 5 & -4 & 2 \\ 2 & 2 & 8\end{bmatrix}$, then adj. (A) equals:

Determinants

Solution:

$A=\begin{bmatrix}4 & -5 & -2 \\ 5 & -4 & 2 \\ 2 & 2 & 8\end{bmatrix}$
$C_{11}=(-1)^{2}\begin{vmatrix}-4 & 2 \\ 2 & 8\end{vmatrix}=-32-4=-36$
$C_{12}=(-1)^{3} \begin{vmatrix}5 & 2 \\ 2 & 8\end{vmatrix}|=-(40-4)=-36$
$C_{13}=(-1)^{4}\begin{vmatrix}5 & -4 \\ 2 & 2\end{vmatrix}=10+8=18$
$C_{21}=(-1)^{3}\begin{vmatrix}-5 & -2 \\ 2 & 8\end{vmatrix}=-(-40+4)=36$
$C_{22}=(-1)^{4}\begin{vmatrix} 4 & -2 \\ 2 & 8\end{vmatrix}=(32+4)=36$
$C _{23}=(-1)^{5}\begin{vmatrix} 4 & -5 \\ 2 & 2 \end{vmatrix}=-(8+10)=-18\\$
$C _{31}=(-1)^{4} \begin{vmatrix}-5 & -2 \\-4 & 2\end{vmatrix}=-10-8=-18$
$C _{32}=(-1)^{5}\begin{vmatrix}4 & -2 \\5 & 2\end{vmatrix}=-(8+10)=-18 \\$
$C_{33}=(-1)^{6}\begin{vmatrix}4 & -5 \\ 5 & -4\end{vmatrix}=-16+25=9$
$\therefore {adj}\, ( A )=\begin{bmatrix} C _{11} & C _{12} & C _{13} \\ C _{21} & C _{22} & C _{23} \\ C _{31} & C _{32} & C _{33}\end{bmatrix}$
$=\begin{bmatrix}-36 & -36 & 18 \\ 36 & 36 & -18 \\ -18 & -18 & 9\end{bmatrix}'=\begin{bmatrix}-36 & 36 & -18 \\ -36 & 36 & -18 \\ 18 & -18 & 9\end{bmatrix}$