Exercise 1.4.21. An alternative way to compute the matrix product AB is as the sum

where is the ith column of A, is the ith row of B, and the product is a matrix.

1. Provide an example showing the procedure above for a 2×2 matrix.
2. Show that the above procedure gives the correct answer for

Answer: (a) We choose the following 2×2 example matrices, with product as shown:

Using the alternative mechanism above we can also compute the product as

(b) For an mxn matrix A and nxp matrix B, the kth column of A and the kth row of B are as follows:

and their matrix product is

so that we have . If we define then we have

so that C = AB as hypothesized.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.