Answer
$a.\qquad B^{2}$ is not defined.
$ b.\qquad F^{2}=\left[\begin{array}{lll}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right]$
Work Step by Step
If $A$ is an $m\times n$ matrix and $B$ is an $n\times k$ matrix
(so the number of columns of $A$ is the same as the number of rows of $B$),
then the matrix product $AB$ is the $m\times k$ matrix
whose $ij$-entry is the inner product of the $i\mathrm{t}\mathrm{h}$ row of $A$ and the jth column of $B.$
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$a.$
$B$ does not have as many columns as it has rows,
so $BB=B^{2}$ is not defined.
$b.$
$F $ is a 3$\times$3 matrix, so $F^{2}$ is defined
$F^{2}=\left[\begin{array}{lll}
1+0+0 & 0+0+0 & 0+0+0\\
0+0+0 & 0+1+0 & 0+0+0\\
0+0+0 & 0+0+0 & 0+0+1
\end{array}\right]=\left[\begin{array}{lll}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right]$
