Q.Find the adjoint of the following matrix : $A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}$

Find all nine cofactors, then transpose the cofactor matrix.

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adj$(A)$ = transpose of the cofactor matrix of $A$.

For $A=\begin{bmatrix}1&-1&2\\3&0&-2\\1&0&3\end{bmatrix}$, compute each cofactor $C_{ij}=(-1)^{i+j}M_{ij}$.

$C_{11}=+\begin{vmatrix}0&-2\\0&3\end{vmatrix}=0,\quad C_{12}=-\begin{vmatrix}3&-2\\1&3\end{vmatrix}=-(9+2)=-11,\quad C_{13}=+\begin{vmatrix}3&0\\1&0\end{vmatrix}=0.$

$C_{21}=-\begin{vmatrix}-1&2\\0&3\end{vmatrix}=-(-3)=3,\quad C_{22}=+\begin{vmatrix}1&2\\1&3\end{vmatrix}=3-2=1,\quad C_{23}=-\begin{vmatrix}1&-1\\1&0\end{vmatrix}=-(0+1)=-1.$

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