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Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams
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Continuing Exercise 3.2.10: Let A be an m n matrix, and let B be an n p matrix. a. Prove

Chapter 3, Problem 22

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QUESTION:

Continuing Exercise 3.2.10: Let A be an m n matrix, and let B be an n p matrix. a. Prove that rank(AB) rank(A). (Hint: Look at part b of Exercise 3.2.10.) b. Prove that if n = p and B is nonsingular, then rank(AB) = rank(A). c. Prove that rank(AB) rank(B). (Hint: Use part a of Exercise 3.2.10 and Theorem 4.6.) d. Prove that if m = n and A is nonsingular, then rank(AB) = rank(B). e. Prove that if rank(AB) = n, then rank(A) = rank(B) = n.

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QUESTION:

Continuing Exercise 3.2.10: Let A be an m n matrix, and let B be an n p matrix. a. Prove that rank(AB) rank(A). (Hint: Look at part b of Exercise 3.2.10.) b. Prove that if n = p and B is nonsingular, then rank(AB) = rank(A). c. Prove that rank(AB) rank(B). (Hint: Use part a of Exercise 3.2.10 and Theorem 4.6.) d. Prove that if m = n and A is nonsingular, then rank(AB) = rank(B). e. Prove that if rank(AB) = n, then rank(A) = rank(B) = n.

ANSWER:

Step 1 of 5

It is given that,  is an  matrix, and  is an  matrix.

a.

To prove that,

               .

 

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