# How do you multiply the matrices ((2, 1), (3, 0), (7, 4)) with ((2, 4), (5, 6))?

Feb 19, 2016

Multiply rows from the first matrix by columns from the second matrix to get the row-column entry for the product.

#### Explanation:

$A \times B = C$

$\left(\begin{matrix}2 & 1 \\ 3 & 0 \\ 7 & 4\end{matrix}\right) \times \left(\begin{matrix}2 & 4 \\ 5 & 6\end{matrix}\right)$

Row 1 of A: $\left(2 , 1\right)$ times Column 1 of B: $\left(\begin{matrix}2 \\ 4\end{matrix}\right)$
gives Row 1: Column 1 of C: $\left(\begin{matrix}2 \times 2 + 1 \times 5 & \text{?" \\ "?" & "?" \\ "?" & "?}\end{matrix}\right)$

Row 1 of A: $\left(2 , 1\right)$ times Column 2 of B: $\left(\begin{matrix}4 \\ 0\end{matrix}\right)$
gives Row 1: Column 2 of C: $\left(\begin{matrix}2 \times 2 + 1 \times 5 & 2 \times 4 + 1 \times 0 \\ \text{?" & "?" \\ "?" & "?}\end{matrix}\right)$

Row 2 of A: 3,0) times Column 1 of B: $\left(\begin{matrix}2 \\ 4\end{matrix}\right)$
gives Row 1: Column 1 of C: $\left(\begin{matrix}32 \times 2 + 1 \times 5 & 2 \times 4 + 1 \times 0 \\ 3 \times 2 + 0 \times 5 & \text{?" \\ "?" & "?}\end{matrix}\right)$

Continue on in this fashion to get:
$\left(\begin{matrix}2 \times 2 + 1 \times 5 & 2 \times 4 + 1 \times 0 \\ 3 \times 2 + 0 \times 5 & 3 \times 4 + 0 \times \\ 7 \times 2 + 4 \times 5 & 7 \times 4 + 4 \times 0\end{matrix}\right) = \left(\begin{matrix}9 & 8 \\ 6 & 12 \\ 34 & 28\end{matrix}\right)$

As a memory aid think about looking at these problems:
through color(red)("rose col"ored glasses.
i.e. $\textcolor{red}{\text{rows}}$ first then $\textcolor{red}{\text{col}}$umns.