Answer
$Adj\,\mathbf{A} = \begin{bmatrix}1&2&-5\\2&1&-7\\-2&-1&4\end{bmatrix}$
$\mathbf{A}^{-1}=\begin{bmatrix}-\frac{1}{3}&-\frac{2}{3}&\frac{5}{3}\\-\frac{2}{3}&-\frac{1}{3}&\frac{7}{3}\\\frac{2}{3}&\frac{1}{3}&-\frac{4}{3}\end{bmatrix}$
Work Step by Step
Given the Matrix:
$\mathbf{A} = \begin{bmatrix}1&1&3\\-2&2&1\\0&1&1\end{bmatrix}$
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 cofactors of $\mathbf{A}$;
$C_{11}=(+)\begin{vmatrix}2&1\\1&1\end{vmatrix}\,,C_{12}=(-)\begin{vmatrix}-2&1\\0&1\end{vmatrix}\,,C_{13}=(+)\begin{vmatrix}-2&2\\0&1\end{vmatrix}$
$C_{21}=(-)\begin{vmatrix}1&3\\1&1\end{vmatrix}\,,C_{22}=(+)\begin{vmatrix}1&3\\0&1\end{vmatrix}\,,C_{23}=(-)\begin{vmatrix}1&1\\0&1\end{vmatrix}$
$C_{31}=(+)\begin{vmatrix}1&3\\2&1\end{vmatrix}\,,C_{32}=(-)\begin{vmatrix}1&3\\-2&1\end{vmatrix}\,,C_{33}=(+)\begin{vmatrix}1&1\\-2&2\end{vmatrix}$
Finding the Matrix of the cofactors we have,
Matrix of Co-factors =$\begin{bmatrix}1&2&-2\\2&1&-1\\-5&-7&4\end{bmatrix}$
$Adj\,\mathbf{A} = \begin{bmatrix}1&2&-2\\2&1&-1\\-5&-7&4\end{bmatrix}^T= \begin{bmatrix}1&2&-5\\2&1&-7\\-2&-1&4\end{bmatrix}$
$\mathbf{A}^{-1}=\frac{1}{det\mathbf{A}}\times Adj\,\mathbf{A}................theorem\,8$
we first find $det\,\mathbf{A}$ using Co-factor expansion;
$det\,\mathbf{A}=1(2\times1-1\times1)-1(-2\times1-1\times0)+3(-2\times1-2\times0)=-3$
therefore,
$\mathbf{A}^{-1}=\frac{1}{-3}\begin{bmatrix}1&2&-5\\2&1&-7\\-2&-1&4\end{bmatrix}=\begin{bmatrix}-\frac{1}{3}&-\frac{2}{3}&\frac{5}{3}\\-\frac{2}{3}&-\frac{1}{3}&\frac{7}{3}\\\frac{2}{3}&\frac{1}{3}&-\frac{4}{3}\end{bmatrix}$
