1. Let A be a square matrix. Assume that the following statement is true: If B is an invertible matrix then rank BA = rank A. (a) Show that if B is an invertible matrix then rank AB = rank A. Hint: What is rank(AB)" ? (b) Show that for any invertible matrix P, rank(P-'AP) = rank A. (c) If P is invertible then nullity P-AP = nullity A.

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