Question

How do you multiply `(4, 5)(3, 4)` by `(3, 0)(0, 3)`?

Answer 1

`((4, 5),(3, 4)) ((3, 0),(0, 3)) = ((12,15),("9","12"))`

Explanation:

`((4, 5),(3, 4)) ((3, 0),(0, 3))`

Find the first row of the product

Take the first row of `((4, 5),(3, 4))`, and make it vertical in front of `((3, 0),(0, 3))`. (We'll do the same for the second row in a minute.)

It looks like:

`{: (4),(5) :}((3, 0),(0, 3))`

Now multiply times the first column and add to get the first number in the first row of the answer:
`4 xx 3 + 5 xx 0 = 12 + 0 = 12`

Next multiply times the second column and add to get the second number in the first row of the answer:
`4 xx 0 + 5 xx 3 = 0 + 15 = 15`

The first row of the product is: `((12,15))`

A this point we know that the product looks like:

`((4, 5),(3, 4)) ((3, 0),(0, 3)) = ((12,15),("-","-"))`

Find the second row of the product
Find the second row of the product by the same process using the second row of `((4, 5),(3, 4))`

`{: (3),(4) :} ((3, 0),(0, 3))` to get: `9` and `12`

The second row of the product is: `((9,12))`

Write the answer

`((4, 5),(3, 4)) ((3, 0),(0, 3)) = ((12,15),("9","12"))`

Answer 2

The second matrix is just a scalar multiple (`3`) of the identity matrix, so the result is just the same scalar multiple of the first matrix, i.e.

`3((4,5),(3,4))=((12,15),(9,12))`

Explanation:

Jim's explanation is really helpful if you are multiplying any two square matrices, but in this particular example there's a short cut.

`((3,0),(0,3)) = 3((1,0),(0,1))`

So:

`((4,5),(3,4))((3,0),(0,3))`

`=3((4,5),(3,4))((1,0),(0,1))`

`=3((4,5),(3,4))`

`=((12,15),(9,12))`