Here is an example (rather than a formula it is a process).
Here is an example
##((4 5)(3 4)(12)) ((3 1)(0 3))##
Find the first row of the product
Take the first row of ##((4 5)(3 4)(12))## 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)(12)) ((3 1)(0 3)) = ((1219)(–)(–))##
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)(12))##
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)(12)) ((3 1)(0 3)) = ((1219)(915)(–))##
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)(12)) ((3 1)(0 3)) = ((1219)(915)(37))##