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Let A € Mmxn (C) and B ¤ Mnxp (℃). (a) Using the rank-nullity theorem, prove that rank(A) = rank(AT). (b) Prove that rank(AB) < min{rank(A), rank(B)}.

Let A € Mmxn (C) and B ¤ Mnxp (℃). (a) Using the rank-nullity theorem, prove that rank(A) = rank(AT). (b) Prove that rank(AB) < min{rank(A), rank(B)}.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let A e Mmxn (C) and Be MnXP(C). (a) Using the rank-nullity theorem, prove that rank(A) = rank(AT). (b) Prove that rank(AB) < min{rank(A), rank(B)}.
Transcribed Image Text:Let A e Mmxn (C) and Be MnXP(C). (a) Using the rank-nullity theorem, prove that rank(A) = rank(AT). (b) Prove that rank(AB) < min{rank(A), rank(B)}.
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