Given matrices , and , we define the matrix product
04. Matrix multiplication, 1
Recall that a 2-by-2 matrix defines for us a transformation of the plane. Suppose we are given two matrices They each define a transformation of the plane. What happens if we first do the transformation associated to , and then do the transformation associated to ?
We get a new transformation associated to a new matrix, which we call .
How on earth can we remember this formula? Here is a mnemonic. Just like when we act on a vector using a matrix, we can think of the entries of as "multiplying a row of into a column of ". More specifically, to get the th entry of (i.e. th row and th column) we multiply the th row of into the th column of :