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

Jun 8, 2016

Matrix multiplication basically involves multiplying each row of the left matrix by all the columns of the right matrix.

I could show you a general notation for this at the end of the answer, but an example is easier to understand.

Since an $M \times K$ multiplied by a $K \times N$ matrix generates an $M \times N$ matrix, a $2 \times 2$ multiplied by a $2 \times 2$ gives a $\setminus m a t h b f \left(2 \times 2\right)$ matrix.

$\textcolor{b l u e}{\left[\begin{matrix}1 & - 4 \\ 4 & - 1\end{matrix}\right] \left[\begin{matrix}3 & - 2 \\ 0 & - 3\end{matrix}\right]}$

$= \left[\begin{matrix}1 \cdot 3 + \left(- 4 \cdot 0\right) & 1 \cdot - 2 + \left(- 4 \cdot - 3\right) \\ 4 \cdot 3 + \left(- 1 \cdot 0\right) & 4 \cdot - 2 + \left(- 1 \cdot - 3\right)\end{matrix}\right]$

$= \textcolor{b l u e}{\left[\begin{matrix}3 & 10 \\ 12 & - 5\end{matrix}\right]}$

The general notation for that (for $2 \times 2$ matrices for simplicity) is:

$\left[\begin{matrix}\textcolor{b l u e}{{a}_{11}} & \textcolor{b l u e}{{a}_{12}} \\ \textcolor{h i g h l i g h t}{{a}_{21}} & \textcolor{h i g h l i g h t}{{a}_{22}}\end{matrix}\right] \times \left[\begin{matrix}\textcolor{\mathmr{and} a n \ge}{{b}_{11}} & \textcolor{red}{{b}_{12}} \\ \textcolor{\mathmr{and} a n \ge}{{b}_{21}} & \textcolor{red}{{b}_{22}}\end{matrix}\right]$

$= \left[\begin{matrix}\textcolor{b l u e}{{a}_{11}} \textcolor{\mathmr{and} a n \ge}{{b}_{11}} + \textcolor{b l u e}{{a}_{12}} \textcolor{\mathmr{and} a n \ge}{{b}_{21}} & \textcolor{b l u e}{{a}_{11}} \textcolor{red}{{b}_{12}} + \textcolor{b l u e}{{a}_{12}} \textcolor{red}{{b}_{22}} \\ \textcolor{h i g h l i g h t}{{a}_{21}} \textcolor{\mathmr{and} a n \ge}{{b}_{11}} + \textcolor{h i g h l i g h t}{{a}_{22}} \textcolor{\mathmr{and} a n \ge}{{b}_{21}} & \textcolor{h i g h l i g h t}{{a}_{21}} \textcolor{red}{{b}_{12}} + \textcolor{h i g h l i g h t}{{a}_{22}} \textcolor{red}{{b}_{22}}\end{matrix}\right]$