(d) Prove that rank(AB) ≤ rank(A).
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help with part d:
Transcribed Image Text:Let A be an m by n matrix and B be an n by n matrix.
(a) If rank(A) = nullity(A), prove that n is even. (Note: nullity(A) = dim Nul(A).)
(b) Prove that Nul(B) < Nul(AB) (i.e., every vector in Nul(B) is also in Nul(AB)).
(c) Prove that dim Nul(B) ≤ dim Nul(AB).
(d) Prove that rank(AB) ≤ rank(A).
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