The formula for matrix multiplication is as follows. Let A be an n x m matrix and let B be an m x k matrix. The product AB is a n x k matrix whose (i,j)-entry is equal to the sum over an index p that ranges from 1 to m of the products of the (i,p)-entry of A with the (p,j)-entry of B. Another way to say this is that the (i,j)-entry is equal to the dot product (also called the inner product) of the ith row of A with the jth column of B. Another way is to multiply by columns. Denote the ith column of B by B_i. The ith column of AB is AB_i.

Matrix multiplication is not always commutative. For example, |1 0| | 1 1 | | 1 1 | | 1 1 | |1 0| |1 -1| |0 -1| x | 0 1 | = | 0 -1 |. But | 0 1 | x |0 -1| = |0 -1|.

Matrices often represent...