How do you do you multiply a 3x2 matrix by a 2x2 matrix??

Here is an example (rather than a formula, it is a process).

Here is an example

##((4, 5),(3, 4),(1,2)) ((3, 1),(0, 3))##

Find the first row of the product

Take the first row of ##((4, 5),(3, 4),(1,2))##, and make it vertical. (We'll do the same for the second row in a minute. And then for the third row.)

##{: (4),(5) :}((3, 1),(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 1 + 5 xx 3 = 4 + 15 = 19##

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

A this point we know that the product looks like:

##((4, 5),(3, 4),(1,2)) ((3, 1),(0, 3)) = ((12,19),("-","-"),("-","-"))##

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),(1,2))##

Make the second row veritcal, muliptly and add.

##{: (3),(4) :}((3, 1),(0, 3))## gets us ##3xx3+4xx0 = 9## and ##3xx1+4xx3=15##

A this point we know that the product looks like:

##((4, 5),(3, 4),(1,2)) ((3, 1),(0, 3)) = ((12,19),(9,15),("-","-"))##

Find the third row of the product.

##{: (1),(2) :} ((3, 1),(0, 3)) ## to get: ##3+0=3## and ##1+6=7##

Write the answer

##((4, 5),(3, 4),(1,2)) ((3, 1),(0, 3)) = ((12,19),(9,15),(3,7))##