Answer
$\left[\begin{array}{ll}
11 & -1\\
-7 & -3
\end{array}\right]$
Work Step by Step
The product of an $m\times\underline{n}$ matrix $A$ and an $\underline{n}\times p$ matrix $B$
is an $m\times p$ matrix $AB$.
The element in the ith row and $j\mathrm{t}\mathrm{h}$ column of $AB$ is found by
multiplying each element in the ith row of $A$ by the corresponding element in the $j\mathrm{t}\mathrm{h}$ column of $B$
and adding the products.
-----------------
$B$ is a $2\times\underline{2}$ matrix, $C$ is a $\underline{2}\times 2$ matrix
$BC$ exists, and is a $2\times 2$ matrix.
$C$ is a $2\times\underline{2}$ matrix, $B$ is a $\underline{2}\times 2$ matrix
$CB$ exists, and is a $2\times 2$ matrix.
BC and CB have the same order, $2\times 2$, so their sum exists.
$BC=\left[\begin{array}{ll}
5(1)+1(-1) & 5(-1)+1(1)\\
-2(1)-2(-1) & -2(-1)-2(1)
\end{array}\right]=\left[\begin{array}{ll}
4 & -4\\
0 & 0
\end{array}\right]$
$CB=\left[\begin{array}{ll}
1(5)-1(-2) & 1(1)-1(-2)\\
-1(5)+1(-2) & -1(1)+1(-2)
\end{array}\right]=\left[\begin{array}{ll}
7 & 3\\
-7 & -3
\end{array}\right]$
$BC+CB=\left[\begin{array}{ll}
4+7 & -4+3\\
0-7 & 0-3
\end{array}\right]=\left[\begin{array}{ll}
11 & -1\\
-7 & -3
\end{array}\right]$
