Answer
See below
Work Step by Step
Given: $A=\begin{bmatrix}
5 & 8 & 16\\4 & 1 & 8\\-4 & -4& -11
\end{bmatrix}$
We will use Cofactor Expansion Theorem to find the determinant of this matrix.
$\det(A)=\sum^n_{j=1}a_{kj}C_{kj}\\
\det(A)=\sum^n_i=1 a_{ik}C_{ik}$
Then $\rightarrow \det(A)=5\begin{vmatrix}
5 & 8\\-4 &-11
\end{vmatrix}-4\begin{vmatrix}
8 & 16\\-4 &-11
\end{vmatrix}+(-4)5\begin{vmatrix}
8 & 16\\1 &8
\end{vmatrix}\\
=5[2(1.(-11)-8.(-4))-4(8.(-11)-16.(-4)]\\
=9$
By using adjoint method, we have
$A^{-1}=\frac{1}{\det(A)}adj(A)=\frac{1}{\det(A)}C^T$
where $C$ is cofactor matrix.
Obtain:
$adj(A)=\begin{bmatrix}
21&24&48\\12 & 9 & 24\\-12 & -12 &-27
\end{bmatrix}$
Consequently, $A^{-1}=\frac{1}{9}\begin{bmatrix}
21&24&48\\12 & 9 & 24\\-12 & -12 &-27
\end{bmatrix}=\begin{bmatrix}
\frac{7}{3}& \frac{8}{3}&\frac{16}{3} \\\frac{4}{3}& 1 & \frac{8}{3} \\\frac{-4}{3}&\frac{-4}{3}& -3
\end{bmatrix}$
