Let A and B be n × n matrices. (a) Show that AB = O if and only if the column space of B is a subspace of the null space of A. (b) Show that if AB = O, then the sum of the ranks of A and B cannot exceed n.

Let A and B be n × n matrices. (a) Show that AB = O if and only if the column space of B is a subspace of the null space of A. (b) Show that if AB = O, then the sum of the ranks of A and B cannot exceed n.

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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Let A and B be n × n matrices.
(a) Show that AB = O if and only if the column
space of B is a subspace of the null space of A.
(b) Show that if AB = O, then the sum of the ranks
of A and B cannot exceed n.

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