# Question #48675

Jul 9, 2015

I will explain how to multiply matrices. (If that's not what you meant by the multiplication law, I apologize for misunderstanding.)

#### Explanation:

There is some information on Multiplication of Matrices here on Socratic.

I think of it as a process that is easier to explain in person, but I'll do my best here.

Let's go through an example:

$\left(\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right) \left(\begin{matrix}3 & 5 \\ 7 & 11\end{matrix}\right)$

Find the first row of the product

Take the first row of $\left(\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right)$, and make it vertical in front of $\left(\begin{matrix}3 & 5 \\ 7 & 11\end{matrix}\right)$. (We'll do the same for the second row in a minute.)

It looks like:

$\left.\begin{matrix}1 \\ 2\end{matrix}\right. \left(\begin{matrix}3 & 5 \\ 7 & 11\end{matrix}\right)$

Now multiply times the first column and add to get the first number in the first row of the answer:
$\left.\begin{matrix}1 \times 3 \\ 2 \times 7\end{matrix}\right. = \left.\begin{matrix}3 \\ 14\end{matrix}\right.$ now add to get $17$

The product starts with:
$\left(\begin{matrix}17 & \text{-" \\ "-" & "-}\end{matrix}\right)$

Next multiply times the second column and add to get the second number in the first row of the answer:
$\left.\begin{matrix}1 \times 5 \\ 2 \times 11\end{matrix}\right. = \left.\begin{matrix}5 \\ 22\end{matrix}\right.$ now add to get $27$

The first row of the product is: $\left(\left(17 , 27\right)\right)$

A this point we know that the product looks like:

$\left(\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right) \left(\begin{matrix}3 & 5 \\ 7 & 11\end{matrix}\right) = \left(\begin{matrix}17 & 27 \\ \text{-" & "-}\end{matrix}\right)$

Find the second row of the product
Find the second row of the product by the same process using the second row of $\left(\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right)$

$\left.\begin{matrix}3 \\ 4\end{matrix}\right. \left(\begin{matrix}3 & 5 \\ 7 & 11\end{matrix}\right)$ to get: $9 + 28 = 37$ and $15 + 44 = 59$

The second row of the product is: $\left(\left(37 , 59\right)\right)$

$\left(\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right) \left(\begin{matrix}3 & 5 \\ 7 & 11\end{matrix}\right) = \left(\begin{matrix}17 & 27 \\ 37 & 59\end{matrix}\right)$