Answer
$B=A^{-1}$
Work Step by Step
We are given the matrices:
$A=\begin{bmatrix}1&2&2\\2&3&3\\1&-1&-1\end{bmatrix}$
$B=\begin{bmatrix}-3&2&0\\7&-4&1\\-5&3&-1\end{bmatrix}$
We have to check is $AB=BA=I_3$.
Compute $AB$:
$AB=\begin{bmatrix}1&2&2\\2&3&3\\1&-1&-1\end{bmatrix}\begin{bmatrix}-3&2&0\\7&-4&1\\-5&3&-1\end{bmatrix}$
$=\begin{bmatrix}-3+14-10&2-8+6&0+2-2\\-6+21-15&4-12+9&0+3-3\\-3-7+10&2+4-6&0-1+2\end{bmatrix}$
$=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
Compute $BA$:
$BA=\begin{bmatrix}-3&2&0\\7&-4&1\\-5&3&-1\end{bmatrix}\begin{bmatrix}1&2&2\\2&3&3\\1&-1&-1\end{bmatrix}$
$=\begin{bmatrix}-3+4+0&-6+6+0&-6+6+0\\7-8+1&14-12-1&14-12-2\\-5+6-1&-10+9+1&-10+9+2\end{bmatrix}$
$=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
Because $AB=BA=I_3$, it means that $B=A^{-1}$.
