Question 10. (5 pts)
Now we will consider a generic example of what you observed in Questions 8 and 9:
Let A be an $M \times N$ matrix, and B be an $N \times M$ matrix. Normally, the best we could say about Rank(AB)
is that it is at most the min \{Rank(A), Rank(B)\}.
However, if we further know that AB is invertible, then we can describe Rank(A), Rank(B), and Rank(AB)
exactly.
In other words, using only the facts that $A \in \mathbb{R}^{M \times N}$, $B \in \mathbb{R}^{N \times M}$, and AB is invertible, prove the values
for Rank(A), Rank(B), and Rank(AB).
(Hint: If you are stumped, start with the invertibility of AB and list all of the things you know about
that)
