Answer
(a) $$AB=\left[ \begin {array}{ccc} -8&-2&-5\\ 4&8&17
\\ -20&1&4\end {array} \right].$$
(a) $$BA=\left[ \begin {array}{ccc} 9&5&4\\3&11&-5
\\ -17&-1&-16\end {array} \right]. $$
Work Step by Step
Given $$
A=\left[\begin{array}{rrr}{2} & {-1} & {3} \\ {5} & {1} & {-2} \\ {2} & {2} & {3}\end{array}\right], \quad B=\left[\begin{array}{rrr}{0} & {1} & {2} \\ {-4} & {1} & {3} \\ {-4} & {-1} & {-2}\end{array}\right]
$$
We have
(a) \begin{align*}
AB&=\left[\begin{array}{rrr}{2} & {-1} & {3} \\ {5} & {1} & {-2} \\ {2} & {2} & {3}\end{array}\right]\left[\begin{array}{rrr}{0} & {1} & {2} \\ {-4} & {1} & {3} \\ {-4} & {-1} & {-2}\end{array}\right]\\
&=\left[ \begin {array}{ccc} -8&-2&-5\\ 4&8&17
\\ -20&1&4\end {array} \right].
\end{align*}
(b) \begin{align*}
BA&=\left[\begin{array}{rrr}{0} & {1} & {2} \\ {-4} & {1} & {3} \\ {-4} & {-1} & {-2}\end{array}\right]\left[\begin{array}{rrr}{2} & {-1} & {3} \\ {5} & {1} & {-2} \\ {2} & {2} & {3}\end{array}\right]\\
&=\left[ \begin {array}{ccc} 9&5&4\\3&11&-5
\\ -17&-1&-16\end {array} \right].
\end{align*}
