# How do you multiply ((2, 3), (4, 5)) with ((27, 12), (47, 22))?

Jul 10, 2016

matrix AXB
195 90
343 158

#### Explanation:

use EXCEL matrix.

matrix A matrix B
2 3 27 12
4 5 47 22
matrix AXB
195 90
343 158

Here is what it would actually look like in Excel:

Notice that if you are not familiar with working with matrix functions in Excel, the method required to define the output range (so more than the top left value is displayed) is not intuitively obvious.

If you use OpenOffice Calc:
Enter the function: =MMULT(B1:C2;B4:C5) followed by Ctrl+Shft+Enter (not just Enter);
also notice the use of the semicolon instead of the comma used by Excel.

Jul 10, 2016

$\left(\begin{matrix}2 & 3 \\ 4 & 5\end{matrix}\right) \times \left(\begin{matrix}27 & 12 \\ 47 & 22\end{matrix}\right) = \textcolor{b r o w n}{\left(\begin{matrix}195 & 90 \\ 343 & 158\end{matrix}\right)}$

#### Explanation:

If $A$, $B$, and $M$ are matrices such that
$\textcolor{w h i t e}{\text{XXX}} A \times B = M$
then the value of the entry in row $i$, column $j$ of $M$ (which we will write as ${M}_{i j}$
is the dot product of row $i$ of $A$ times column $j$ of $B$

For the given example

$\left(\begin{matrix}\textcolor{red}{2} & \textcolor{red}{3} \\ \textcolor{b l u e}{4} & \textcolor{b l u e}{5}\end{matrix}\right) \times \left(\begin{matrix}\textcolor{g r e e n}{27} & \textcolor{p u r p \le}{12} \\ \textcolor{g r e e n}{47} & \textcolor{p u r p \le}{22}\end{matrix}\right)$)

$= \left(\left(\textcolor{red}{2} \times \textcolor{g r e e n}{27} + \textcolor{red}{3} \times \textcolor{g r e e n}{47} , \textcolor{w h i t e}{\text{XXX"),color(red)(2)xxcolor(purple)(12)+color(red)(3)xxcolor(purple)(22)), (color(blue)(4)xxcolor(green)(27)+color(blue)(5)xxcolor(green)(47),color(white)("XXX}} , \textcolor{b l u e}{4} \times \textcolor{p u r p \le}{12} + \textcolor{b l u e}{5} \times \textcolor{p u r p \le}{22}\right)\right)$

$= \left(\begin{matrix}195 & 90 \\ 343 & 158\end{matrix}\right)$

Remember the row, column entry in your answer is the calculated using:
the row from the first matrix, and
the column from the second matrix.