4. Let A be a 3 x 4 matrix and B be a 4 x 5 matrix such that ABx = 0 for all x E R°. a. Show that R(B) C N(A) and deduce that rank(B) < null(A) b. Use the Rank-Nullity theorem to prove that rank(A) + rank(B) < 4.

4. Let A be a 3 x 4 matrix and B be a 4 x 5 matrix such that ABx = 0 for all x E R°. a. Show that R(B) C N(A) and deduce that rank(B) < null(A) b. Use the Rank-Nullity theorem to prove that rank(A) + rank(B) < 4.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 77E: Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and...

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Question

4. Let A be a 3 x 4 matrix and B be a 4 x 5 matrix such that ABx = 0 for all x E R°.
a. Show that R(B) C N(A) and deduce that rank(B) < null(A)
b. Use the Rank-Nullity theorem to prove that rank(A) + rank(B) < 4.

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