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(d) Prove that rank(AB) ≤ rank(A).

(d) Prove that rank(AB) ≤ rank(A).

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Please help with part d:

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).
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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