Question

How do you multiply `[(1,2,4), (-1,3,0)]*[(2,-4), (3,5), (-1,0)]`?

Answer 1

Here is the method I was taught. (There is a dual method that is also useful.)

Explanation:

`((1,2,4),(-1,3,0)) ((2,-4),(3,5),(-1,0))`

Find the first row of the product

Take the first row of `((1,2,4),(-1,3,0))`, and make it vertical. (We'll do the same for the second row in a minute.)

`{: (1),(2),(4) :}((2,-4),(3,5),(-1,0))`

Now multiply times the first column and add to get the first number in the first row of the answer:
`1 xx 2 + 2 xx 3 + 4 xx -1 = 2+6+(-4) = 4`

Next multiply times the second column and add to get the second number in the first row of the answer:
`1 xx -4 + 2 xx 5 + 4 xx 0 = -4+10+0 = 6`

(If there were more columns in the second matrix, we would continue this process.)

A this point we know that the product looks like:

`((1,2,4),(-1,3,0)) ((2,-4),(3,5),(-1,0)) = ((4,6),("-","-"))`

Find the second row of the product

Find the second row of the product by the same process using the second row of `((1,2,4),(-1,3,0))`

Make the second row vertical, multiply and add.

`{: (-1),(3),(0) :}((2,-4),(3,5),(-1,0))`

gets us `-1xx2+3xx3+0xx(-1) = -2+9+0=7`

and `-1xx-4+3xx5+0xx0 = 4+15+0=19`

So the product is:

`((1,2,4),(-1,3,0)) ((2,-4),(3,5),(-1,0)) = ((4,6),(7,19))`

Note
If there were another row in the first matrix, then we would find a third row for the product.

Answer 2

This method is the dual of the one I used in the other answer.

Explanation:

`((1,2,4),(-1,3,0)) ((2,-4),(3,5),(-1,0))`

Find the first column of the product

Take the first column of the second matrix. Use its entries to build a linear combination of the columns of the first matrix.
(We'll do the same for the second column in a minute.)

First column of second matrix: `((2),(3),(-1))`

Linear combination of the columns of the first matrix:

`2((1),(-1)) + 3((2),(3))+(-1)((4),(0)) = ((2+6-4),(-2+9+0)) = ((4),(7))`

this is the first column of the product.

A this point we know that the product looks like:

`((1,2,4),(-1,3,0)) ((2,-4),(3,5),(-1,0)) = ((4,"-"),(7,"-"))`

Find the second column of the product

Find the second column of the product by the same process using the values in second column of `((2,-4),(3,5),(-1,0))` as coefficients for a linear combination of the columns in the first matrix.

Second column of second matrix: `((-4),(5),(0))`

Linear combination of the columns of the first matrix:

`-4((1),(-1)) + 5((2),(3))+0((4),(0)) = ((-4+10+0),(4+15+0)) = ((6),(19))`

this is the second column of the product.

So the product is:

`((1,2,4),(-1,3,0)) ((2,-4),(3,5),(-1,0)) = ((4,6),(7,19))`