Answer
False.
Work Step by Step
Counterexample: $A=\left[\begin{array}{ll}
1 & 0\\
1 & 0
\end{array}\right],B=\left[\begin{array}{ll}
0 & 0\\
1 & 1
\end{array}\right]$
$AB=\left[\begin{array}{ll}
0 & 0\\
0 & 0
\end{array}\right]$
$BA=\left[\begin{array}{ll}
0 & 0\\
2 & 0
\end{array}\right]$
Not all corresponding entries of AB and BA are equal, so
$AB\neq BA$
The statement is false.
