Answer
(a) $$AB=\left[\begin{array}{rrr}{0} & {15} \\ {6} & {12} \end{array}\right].$$
(b) $$BA=\left[\begin{array}{rrr}{-2} & {2} \\ {31} & {14} \end{array}\right].$$
Work Step by Step
Given $$
A=\left[\begin{array}{rrr}{1} & {2} \\ {4} & {2} \end{array}\right], \quad \text { and } \quad B=\left[\begin{array}{rrr}{2} & {-1} \\ {-1} & {8} \end{array}\right].
$$
We have
(a) \begin{align*}
AB&=\left[\begin{array}{rrr}{1} & {2} \\ {4} & {2} \end{array}\right] \left[\begin{array}{rrr}{2} & {-1} \\ {-1} & {8} \end{array}\right]\\
&=\left[\begin{array}{rrr}{2-2} & {-1+16} \\ {8-2} & {-4+16} \end{array}\right]\\
&=\left[\begin{array}{rrr}{0} & {15} \\ {6} & {12} \end{array}\right].
\end{align*}
(b) \begin{align*}
BA&=\left[\begin{array}{rrr}{2} & {-1} \\ {-1} & {8} \end{array}\right]\left[\begin{array}{rrr}{1} & {2} \\ {4} & {2} \end{array}\right] \\
&=\left[\begin{array}{rrr}{2-4} & {4-2} \\ {-1+32} & {-2+16} \end{array}\right]\\
&=\left[\begin{array}{rrr}{-2} & {2} \\ {31} & {14} \end{array}\right].
\end{align*}