Chapter 5.4, Problem 24E

Verify the statements in Exercises 19–24. The matrices are square.

24. If A and B are similar, then they have the same rank. [Hint: Refer to Supplementary Exercises 13 and 14 for Chapter 4.]

Refer to Chapter 4

13. Show that if P is an invertible m × m matrix, then rank PA = rank A. [Hint: Apply Exercise 12 to PA and P⁻¹(PA).]

14. Show that if Q is invertible, then rank AQ = rank A. [Hint: Use Exercise 13 to study rank(AQ)^T.]