Answer
$Adj\,\mathbf{A} = \begin{bmatrix}-1&0&0\\-1&-5&0\\-1&-15&5\end{bmatrix}$
$\mathbf{A}^{-1}=\frac{1}{-5}\begin{bmatrix}-1&0&0\\-1&-5&0\\-1&-15&5\end{bmatrix}=\begin{bmatrix}0.2&0&0\\0.2&1&0\\0.2&3&-1 \end{bmatrix}$
Work Step by Step
Given the Matrix:
$\mathbf{A} = \begin{bmatrix}5&0&0\\-1&1&0\\-2&3&-1\end{bmatrix}$
Using the following formula to find the adjugate;
$Adj\,\mathbf{A}=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\C_{21}&C_{22}&C_{23}\\C_{31}&C_{32}&C_{33}\end{bmatrix}^T$
$C_{ij}$ are the Matrix of minors;
$C_{11}=(+)\begin{vmatrix}1&0\\3&-1\end{vmatrix}\,,C_{12}=(-)\begin{vmatrix}-1&0\\-2&-1\end{vmatrix}\,,C_{13}=(+)\begin{vmatrix}-1&1\\-2&3\end{vmatrix}$
$C_{21}=(-)\begin{vmatrix}0&0\\3&-1\end{vmatrix}\,,C_{22}=(+)\begin{vmatrix}5&0\\-2&-1\end{vmatrix}\,,C_{23}=(-)\begin{vmatrix}5&0\\-2&3\end{vmatrix}$
$C_{31}=(+)\begin{vmatrix}0&0\\1&0\end{vmatrix}\,,C_{32}=(-)\begin{vmatrix}5&0\\-1&0\end{vmatrix}\,,C_{33}=(+)\begin{vmatrix}5&0\\-1&1\end{vmatrix}$
Finding the Matrix of the cofactors we have,
Matrix of Co-factors =$\begin{bmatrix}-1&-1&-1\\0&-5&-15\\0&0&5\end{bmatrix}$
$Adj\,\mathbf{A} =\begin{bmatrix}-1&-1&-1\\0&-5&-15\\0&0&5\end{bmatrix}^T= \begin{bmatrix}-1&0&0\\-1&-5&0\\-1&-15&5\end{bmatrix}$
$\mathbf{A}^{-1}=\frac{1}{det\mathbf{A}}\times Adj\,\mathbf{A}................theorem\,8$
we first find $det\,\mathbf{A}$ using Co-factor expansion;
$det\,\mathbf{A}=5(-1\times1-0\times3)+0+0=-5$
therefore,
$\mathbf{A}^{-1}=\frac{1}{-5}\begin{bmatrix}-1&0&0\\-1&-5&0\\-1&-15&5\end{bmatrix}=\begin{bmatrix}0.2&0&0\\0.2&1&0\\0.2&3&-1 \end{bmatrix}$
