Exercise 2.4.7. If is an by matrix with rank answer the following:

a) When is invertible, with existing such that ?

b) When does have an infinite number of solutions for any ?

Answer: a) Per theorem 2Q on page 96, for to have a right inverse such that we must have and . Per the same theorem, for to have a left inverse such that we must have and . For to have both a left and right inverse we must therefore have , in which case .

b) Per theorem 2Q on page 96, for to have at least one solution we must have . For to have an infinite number of solutions there must be at least one free variable, i.e., the rank must be less than the number of columns . We therefore must have and .

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.