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

2 Answers
Apr 14, 2017

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.

Apr 14, 2017

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))#