# How do you multiply ((2, 1), (-1, 0)) by ((4, 2), (3, -1))?

Aug 15, 2016

$\left(\begin{matrix}2 & 1 \\ - 1 & 0\end{matrix}\right) \times \left(\begin{matrix}4 & 2 \\ 3 & - 1\end{matrix}\right) = \left(\begin{matrix}\textcolor{red}{11} & \textcolor{b l u e}{3} \\ \textcolor{g r e e n}{- 4} & \textcolor{m a \ge n t a}{- 2}\end{matrix}\right)$

#### Explanation:

1. Check if they are compatible.

A 2x2 matrix times a 2x2 matrix will give a 2x2 matrix.

Each row (R) in the first matrix is multiplied by each column (C) in the second matrix.

$\left(\begin{matrix}2 & 1 \\ - 1 & 0\end{matrix}\right) \times \left(\begin{matrix}4 & 2 \\ 3 & - 1\end{matrix}\right) = \left(\begin{matrix}\textcolor{red}{a} & \textcolor{b l u e}{b} \\ \textcolor{g r e e n}{c} & \textcolor{m a \ge n t a}{d}\end{matrix}\right)$

$\textcolor{red}{a} = {R}_{1} \times {C}_{1} \Rightarrow = 2 \times 4 + 1 \times 3 = \textcolor{red}{11}$
$\textcolor{b l u e}{b} = {R}_{1} \times {C}_{2} \Rightarrow = 2 \times 2 + 1 \times \left(- 1\right) = \textcolor{b l u e}{3}$
$\textcolor{g r e e n}{c} = {R}_{2} \times {C}_{1} \Rightarrow = - 1 \times 4 + 0 \times 3 = \textcolor{g r e e n}{- 4}$
$\textcolor{m a \ge n t a}{d} = {R}_{2} \times {C}_{2} \Rightarrow = - 1 \times 2 + 0 \times \left(- 1\right) = \textcolor{m a \ge n t a}{- 2}$

$\left(\begin{matrix}2 & 1 \\ - 1 & 0\end{matrix}\right) \times \left(\begin{matrix}4 & 2 \\ 3 & - 1\end{matrix}\right) = \left(\begin{matrix}\textcolor{red}{11} & \textcolor{b l u e}{3} \\ \textcolor{g r e e n}{- 4} & \textcolor{m a \ge n t a}{- 2}\end{matrix}\right)$