Review exercise 2.18. Suppose that is an by matrix with rank . Show that if then .

Answer: Since the rank of is we know that the columns of are linearly independent and that the inverse exists. We then have

So if and the rank then .

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.