Answer
$Adj\,\mathbf{A} = \begin{bmatrix}6&4&14\\0&-2&-1\\0&0&-3\end{bmatrix}$
$\mathbf{A}^{-1}=\frac{1}{6}\begin{bmatrix}6&4&14\\0&-2&-1\\0&0&-3\end{bmatrix}=\begin{bmatrix}1&\frac{2}{3}&\frac{7}{3}\\0&-\frac{1}{3}&-\frac{1}{6}\\0&0&-\frac{1}{2}\end{bmatrix}$
Work Step by Step
Given the Matrix:
$\mathbf{A} = \begin{bmatrix}1&2&4\\0&-3&1\\0&0&-2\end{bmatrix}$
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;
$C_{11}=(+)\begin{vmatrix}-3&1\\0&-2\end{vmatrix}\,,C_{12}=(-)\begin{vmatrix}0&1\\0&-2\end{vmatrix}\,,C_{13}=(+)\begin{vmatrix}0&-3\\0&0\end{vmatrix}$
$C_{21}=(-)\begin{vmatrix}2&4\\0&-2\end{vmatrix}\,,C_{22}=(+)\begin{vmatrix}1&4\\0&-2\end{vmatrix}\,,C_{23}=(-)\begin{vmatrix}1&2\\0&0\end{vmatrix}$
$C_{31}=(+)\begin{vmatrix}2&4\\-3&1\end{vmatrix}\,,C_{32}=(-)\begin{vmatrix}1&4\\0&1\end{vmatrix}\,,C_{33}=(+)\begin{vmatrix}1&2\\0&-3\end{vmatrix}$
Finding the Matrix of the cofactors we have,
Matrix of Co-factors =$\begin{bmatrix}6&0&0\\4&-2&0\\14&-1&-3\end{bmatrix}$
$Adj\,\mathbf{A} = \begin{bmatrix}6&0&0\\4&-2&0\\14&-1&-3\end{bmatrix}^T=\begin{bmatrix}6&4&14\\0&-2&-1\\0&0&-3\end{bmatrix}$
$\mathbf{A}^{-1}=\frac{1}{det\mathbf{A}}\times Adj\,\mathbf{A}................theorem\,8$
we first find $det\,\mathbf{A}$ using Co-factor expansion;
$det\,\mathbf{A}=1(3\times2-1\times0)-2(-2\times0-1\times0)+4(0+0)=6$
therefore,
$\mathbf{A}^{-1}=\frac{1}{6}\begin{bmatrix}6&4&14\\0&-2&-1\\0&0&-3\end{bmatrix}=\begin{bmatrix}1&\frac{2}{3}&\frac{7}{3}\\0&-\frac{1}{3}&-\frac{1}{6}\\0&0&-\frac{1}{2}\end{bmatrix}$
