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

Dec 21, 2017

$\left(\begin{matrix}1 & - 3 & 2 \\ 2 & 1 & - 3 \\ 4 & - 3 & - 1\end{matrix}\right) \left(\begin{matrix}2 & 1 & - 1 & - 2 \\ 3 & - 2 & - 1 & - 1 \\ 2 & - 5 & - 1 & 0\end{matrix}\right) = \left(\begin{matrix}- 3 & - 3 & 0 & 1 \\ 1 & 15 & 0 & - 5 \\ - 3 & 15 & 0 & - 5\end{matrix}\right)$

#### Explanation:

For each row of the first matrix and column of the second, sum the products of the three corresponding terms...

((color(red)(1), color(red)(-3), color(red)(2)), (2, 1, -3), (4, -3, -1))((color(blue)(2), 1, -1, -2),(color(blue)(3), -2, -1, -1),(color(blue)(2), -5, -1, 0)) = ((color(purple)(-3), ?, ?, ?),(?, ?, ?, ?), (?, ?, ?, ?))

$\left(\textcolor{red}{1}\right) \left(\textcolor{b l u e}{2}\right) + \left(\textcolor{red}{- 3}\right) \left(\textcolor{b l u e}{3}\right) + \left(\textcolor{red}{2}\right) \left(\textcolor{b l u e}{2}\right) = 2 - 9 + 4 = \textcolor{p u r p \le}{- 3}$

((color(red)(1), color(red)(-3), color(red)(2)), (2, 1, -3), (4, -3, -1))((2, color(blue)(1), -1, -2),(3, color(blue)(-2), -1, -1),(2, color(blue)(-5), -1, 0)) = ((-3, color(purple)(-3), ?, ?),(?, ?, ?, ?), (?, ?, ?, ?))

$\left(\textcolor{red}{1}\right) \left(\textcolor{b l u e}{1}\right) + \left(\textcolor{red}{- 3}\right) \left(\textcolor{b l u e}{- 2}\right) + \left(\textcolor{red}{2}\right) \left(\textcolor{b l u e}{- 5}\right) = 1 + 6 - 10 = \textcolor{p u r p \le}{- 3}$

dot dot dot

((1, -3, 2), (color(red)(2), color(red)(1), color(red)(-3)), (4, -3, -1))((color(blue)(2), 1, -1, -2),(color(blue)(3), -2, -1, -1),(color(blue)(2), -5, -1, 0)) = ((-3, -3, 0, 1),(color(purple)(1), ?, ?, ?), (?, ?, ?, ?))

$\left(\textcolor{red}{2}\right) \left(\textcolor{b l u e}{2}\right) + \left(\textcolor{red}{1}\right) \left(\textcolor{b l u e}{3}\right) + \left(\textcolor{red}{- 3}\right) \left(\textcolor{b l u e}{2}\right) = 4 + 3 - 6 = \textcolor{p u r p \le}{1}$

dot dot dot

$\left(\begin{matrix}1 & - 3 & 2 \\ 2 & 1 & - 3 \\ \textcolor{red}{4} & \textcolor{red}{- 3} & \textcolor{red}{- 1}\end{matrix}\right) \left(\begin{matrix}2 & 1 & - 1 & \textcolor{b l u e}{- 2} \\ 3 & - 2 & - 1 & \textcolor{b l u e}{- 1} \\ 2 & - 5 & - 1 & \textcolor{b l u e}{0}\end{matrix}\right) = \left(\begin{matrix}- 3 & - 3 & 0 & 1 \\ 1 & 15 & 0 & - 5 \\ - 3 & 15 & 0 & \textcolor{p u r p \le}{- 5}\end{matrix}\right)$

$\left(\textcolor{red}{4}\right) \left(\textcolor{b l u e}{- 2}\right) + \left(\textcolor{red}{- 3}\right) \left(\textcolor{b l u e}{- 1}\right) + \left(\textcolor{red}{- 1}\right) \left(\textcolor{b l u e}{0}\right) = - 8 + 3 + 0 = \textcolor{p u r p \le}{- 5}$