Exercise 1.6.23. Assume that and are square matrices, and that is invertible. Show that is invertible as well. (Use the fact that .)
Answer: First, since and are square matrices we know that both of the product matrices and exist and have the same number of rows and columns. We then have
Since we are assuming that the inverse of exists, we have
Multiplying both sides of the resulting equation on the left by and then adding to both sides, we have
So is a left inverse for . We then multiply by on the right:
So is also a right inverse for . Since is both a left inverse and right inverse for we conclude that is invertible (with as its inverse).
We have thus showed that if is invertible then is also invertible.
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.