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

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Answer 1 · 2015-10-22T22:37:33

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

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