Review exercise 1.22. Answer the following questions:
(a) If has an inverse, does also have an inverse? If so, what is it?
(b) If is both invertible and symmetric, what is the transpose of ?
(c) Illustrate (a) and (b) when .
Answer: (a) Assume is invertible, so that exists. Then we have and therefore . But so we have and is a left inverse for .
Similarly we have and therefore . But so we have and is a right inverse for .
Since is both a left inverse and a right inverse for we see that is invertible and .
(b) If is symmetric then we have . If is also invertible then exists and from (a) above we know that . We then have . Since we see that is also symmetric if is.
(c) If we have
Note that is symmetric just as is.
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.