Exercise 2.5.14. Suppose we have the block matrices

with the following properties:

- and have rows
- and have rows
- and have rows
- and have rows
- and are square matrices

What is the product of the two matrices, and what are the sizes of each of the product’s component submatrices?

Answer: Since is a square matrix with rows it must have columns. If has columns then must have columns also. Similarly, since is a square matrix with rows it must have columns, and must have columns as well.

In computing the (1, 1) element of the product matrix we multiply by and by . (See the post on multiplying block matrices.) This means that the number of columns of must equal the number of rows of or , and the number of columns of must equal the number of rows of or . The number of columns of and must then be and respectively.

The sizes of each of the elements through is thus as follows:

- is by
- is by
- is by
- is by
- is by
- is by
- is by
- is by

The product matrix is thus

with the size of each element as follows:

- is by
- is by
- is by
- is by

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.