Answer
$AB=BA={{I}_{3}}$
Work Step by Step
Consider the given matrices,
$\begin{align}
& AB=\left[ \begin{array}{*{35}{r}}
1 & 2 & 2 \\
2 & 3 & 3 \\
1 & -1 & -2 \\
\end{array} \right]\left[ \begin{array}{*{35}{r}}
-3 & 2 & 0 \\
7 & -4 & 1 \\
-5 & 3 & -1 \\
\end{array} \right] \\
& =\left[ \begin{array}{*{35}{r}}
-3+14-10 & 2-8+6 & 0+2-2 \\
-6+21-15 & 4-12+9 & 0+3-3 \\
-3-7+10 & 2+4-6 & 0-1+2 \\
\end{array} \right] \\
& =\left[ \begin{array}{*{35}{r}}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} \right]
\end{align}$
Consider,
$\begin{align}
& BA=\left[ \begin{array}{*{35}{r}}
-3 & 2 & 0 \\
7 & -4 & 1 \\
-5 & 3 & -1 \\
\end{array} \right]\left[ \begin{array}{*{35}{r}}
1 & 2 & 2 \\
2 & 3 & 3 \\
1 & -1 & -2 \\
\end{array} \right] \\
& =\left[ \begin{array}{*{35}{r}}
-3+4+0 & -6+6+0 & -6+6+0 \\
7-8+1 & 14-12-1 & 14-12-2 \\
-5+6-1 & -10+9+1 & -10+9+2 \\
\end{array} \right] \\
& =\left[ \begin{array}{*{35}{r}}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} \right]
\end{align}$
Therefore
$AB=BA={{I}_{3}}$
Hence, by the definition of the inverse, the matrix $B$ is the inverse of the matrix $A$.
