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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)

          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)

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

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