Answer
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1
\\ -2&2&1\end {array} \right]
.$$
$$A^{-1}= -\frac{1}{6}\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1
\\ -2&2&1\end {array} \right].$$
Work Step by Step
The matrix is given by $A=\left[ \begin {array}{ccc} 1&2&3\\ 0&1&-1
\\ 2&2&2\end {array} \right]
.$ To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows
$$\left[ \begin {array}{ccc} 4&-2&-2\\ 2&-4&2
\\ -5&1&1\end {array} \right]
.$$
Now, the adjoint of $A$ is
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1
\\ -2&2&1\end {array} \right]
.$$
To find $A^{-1}$, we have to calculate $\det(A)$ which is given by
$$\det(A)=-6.$$
Finally, we have
$$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)=-\frac{1}{6}\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1
\\ -2&2&1\end {array} \right].$$
