# How do you multiply ((3, 1, -2, 0), (1, 4, -2, -2), (-2, 0, 7, -3), (-1, 3, -1, 2)) with ((1, 1), (1, -2), (-3, 2), (4, 5))?

Mar 23, 2018

$\left(\begin{matrix}3 & 1 & - 2 & 0 \\ 1 & 4 & - 2 & - 2 \\ - 2 & 0 & 7 & - 3 \\ - 1 & 3 & - 1 & 2\end{matrix}\right) \left(\begin{matrix}1 & 1 \\ 1 & - 2 \\ - 3 & 2 \\ 4 & 5\end{matrix}\right) = \left(\begin{matrix}10 & - 3 \\ 3 & - 21 \\ - 35 & - 3 \\ 13 & 1\end{matrix}\right)$

#### Explanation:

When we multiply matrices, we must first check that they're compatible for multiplication.

$\left(\begin{matrix}3 & 1 & - 2 & 0 \\ 1 & 4 & - 2 & - 2 \\ - 2 & 0 & 7 & - 3 \\ - 1 & 3 & - 1 & 2\end{matrix}\right)$ is $4 \times 4$

$\left(\begin{matrix}1 & 1 \\ 1 & - 2 \\ - 3 & 2 \\ 4 & 5\end{matrix}\right)$ is $4 \times 2$

So, they are compatible and the dimensions of the product are $4 \times 2$. Now, when we multiply matrices we do it by multiplying each row of the first matrix with each column of the second.

So,

$\left(\begin{matrix}3 & 1 & - 2 & 0 \\ 1 & 4 & - 2 & - 2 \\ - 2 & 0 & 7 & - 3 \\ - 1 & 3 & - 1 & 2\end{matrix}\right) \left(\begin{matrix}1 & 1 \\ 1 & - 2 \\ - 3 & 2 \\ 4 & 5\end{matrix}\right) = \left(\begin{matrix}10 & - 3 \\ 3 & - 21 \\ - 35 & - 3 \\ 13 & 1\end{matrix}\right)$