Answer
(a) $$AB=\left[\begin{array}{rrr}{4} & {6} \\ {4} & {-9} \end{array}\right].$$
(b) $$BA=\left[\begin{array}{rrr}{7} & {-4} \\ {6} & {-12} \end{array}\right].$$
Work Step by Step
Given $$
A=\left[\begin{array}{rr}{2} & {-2} \\ {-1} & {4}\end{array}\right], \quad B=\left[\begin{array}{rr}{4} & {1} \\ {2} & {-2}\end{array}\right]
$$
We have
(a) \begin{align*}
AB&=\left[\begin{array}{rr}{2} & {-2} \\ {-1} & {4}\end{array}\right]\left[\begin{array}{rr}{4} & {1} \\ {2} & {-2}\end{array}\right]]\\
&=\left[\begin{array}{rrr}{8-4} & {2+4} \\ {-4+8} & {-1-8} \end{array}\right]\\
&=\left[\begin{array}{rrr}{4} & {6} \\ {4} & {-9} \end{array}\right].
\end{align*}
(b) \begin{align*}
BA&=\left[\begin{array}{rr}{4} & {1} \\ {2} & {-2}\end{array}\right]\left[\begin{array}{rr}{2} & {-2} \\ {-1} & {4}\end{array}\right]\\
&=\left[\begin{array}{rrr}{8-1} & {-8+4} \\ {4+2} & {-4-8} \end{array}\right]\\
&=\left[\begin{array}{rrr}{7} & {-4} \\ {6} & {-12} \end{array}\right].
\end{align*}
