Answer
See below
Work Step by Step
Given: $A=\begin{bmatrix}
2 &6 &6\\2 & 7&6\\2&7&7
\end{bmatrix}$
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 $\rightarrow \det(A)=2\begin{vmatrix}
7 & 6\\7 &7
\end{vmatrix}-2\begin{vmatrix}
6 & 6\\7 &7
\end{vmatrix}+2\begin{vmatrix}
6 & 6\\7 &6
\end{vmatrix}\\
=2(7.7-6.7)-2(6.7-6.7)+2(6.6-6.7)\\
=2$
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:
$adj(A)=\begin{bmatrix}
7 &0 & -6\\-2 & 2 &0\\0&-2&2
\end{bmatrix}$
Consequently, $A^{-1}=\frac{1}{2}\begin{bmatrix}
7 &0 & -6\\-2 & 2 &0\\0&-2&2
\end{bmatrix}=\begin{bmatrix}
\frac{7}{2}& 0& -3\\-1& 1 & 0 \\0 & -1 &1
\end{bmatrix}$
