Q.Find the inverse of the following matrix by the adjoint method. $\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}$

Step 1: $A=\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}$ is lower triangular, so $|A|=1\times3\times(-1)=-3\neq0$.

Step 2: Cofactors: $A_{11}=\begin{vmatrix}3&0\\2&-1\end{vmatrix}=-3$; $A_{12}=-\begin{vmatrix}3&0\\5&-1\end{vmatrix}=-(-3)=3$; $A_{13}=\begin{vmatrix}3&3\\5&2\end{vmatrix}=6-15=-9$; $A_{21}=-\begin{vmatrix}0&0\\2&-1\end{vmatrix}=0$; $A_{22}=\begin{vmatrix}1&0\\5&-1\end{vmatrix}=-1$; $A_{23}=-\begin{vmatrix}1&0\\5&2\end{vmatrix}=-2$; $A_{31}=\begin{vmatrix}0&0\\3&0\end{vmatrix}=0$; $A_{32}=-\begin{vmatrix}1&0\\3&0\end{vmatrix}=0$; $A_{33}=\begin{vmatrix}1&0\\3&3\end{vmatrix}=3$. …

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