Q.Find the inverse of the following matrix by the adjoint method. $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$

Step 1: $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$ is upper triangular, so $|A|=1\times2\times5=10\neq0$.

Step 2: Cofactors: $A_{11}=\begin{vmatrix}2&4\\0&5\end{vmatrix}=10$; $A_{12}=-\begin{vmatrix}0&4\\0&5\end{vmatrix}=0$; $A_{13}=\begin{vmatrix}0&2\\0&0\end{vmatrix}=0$; $A_{21}=-\begin{vmatrix}2&3\\0&5\end{vmatrix}=-10$; $A_{22}=\begin{vmatrix}1&3\\0&5\end{vmatrix}=5$; $A_{23}=-\begin{vmatrix}1&2\\0&0\end{vmatrix}=0$; $A_{31}=\begin{vmatrix}2&3\\2&4\end{vmatrix}=8-6=2$; $A_{32}=-\begin{vmatrix}1&3\\0&4\end{vmatrix}=-4$; $A_{33}=\begin{vmatrix}1&2\\0&2\end{vmatrix}=2$. …

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