Answer
a. $AB=\left[\begin{array}{ll}
-10 & 12\\
25 & -30
\end{array}\right]$
b. $BA=\left[\begin{array}{ll}
0 & 0\\
9 & -40
\end{array}\right]$
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.
-----------------
a.
$A$ is a $2\times\underline{2}$ matrix, B is a $\underline{2}\times 2$ matrix
$AB$ exists, and is a 2$\times$2 matrix.
$AB=\left[\begin{array}{ll}
3(0)+(-2)(5) & 3(0)+(-2)(-6)\\
1(0)+5(5) & 1(0)+5(-6)
\end{array}\right]=\left[\begin{array}{ll}
-10 & 12\\
25 & -30
\end{array}\right]$
b.
$B$ is a $2\times\underline{2}$ matrix, $A$ is a $\underline{2}\times 2$ matrix
$BA$ exists, and is a 2$\times$2 matrix.
$BA=\left[\begin{array}{ll}
0(3)+0(1) & 0(-2)+0(5)\\
5(3)+(-6)(1) & 5(-2)+(-6)(5
\end{array}\right]=\left[\begin{array}{ll}
0 & 0\\
9 & -40
\end{array}\right]$
