Q.Find the inverse of $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix}$ by adjoint method.

Step 1: $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix}$.

Step 2: $A_{11}=\begin{vmatrix}2&3\\2&1\end{vmatrix}=2-6=-4$; $A_{12}=-\begin{vmatrix}0&3\\1&1\end{vmatrix}=-(0-3)=3$; $A_{13}=\begin{vmatrix}0&2\\1&2\end{vmatrix}=0-2=-2$; $A_{21}=-\begin{vmatrix}0&1\\2&1\end{vmatrix}=-(0-2)=2$; $A_{22}=\begin{vmatrix}1&1\\1&1\end{vmatrix}=1-1=0$; $A_{23}=-\begin{vmatrix}1&0\\1&2\end{vmatrix}=-(2-0)=-2$; $A_{31}=\begin{vmatrix}0&1\\2&3\end{vmatrix}=0-2=-2$; $A_{32}=-\begin{vmatrix}1&1\\0&3\end{vmatrix}=-(3-0)=-3$; $A_{33}=\begin{vmatrix}1&0\\0&2\end{vmatrix}=2-0=2$.

Step 3: Cofactor matrix $\begin{bmatrix}-4&3&-2\\2&0&-2\\-2&-3&2\end{bmatrix}$, so $\text{adj}\,A=\begin{bmatrix}-4&2&-2\\3&0&-3\\-2&-2&2\end{bmatrix}$. …

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