Suppose A and B are n × n matrices. 1. Column space C(AB) is a subspace of C(A). 2. rank(AB) ≤ rank(A). (a) If AB = I, s

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Suppose A and B are n × n matrices.
1. Column space C(AB) is a subspace of C(A).
2. rank(AB) ≤ rank(A).
(a) If AB = I, show that A is invertible using the properties 1
and 2, (but not using the existence of a right inverse B, nor
det(AB) = det(A)· det(B).)
(b) If AB = I, show that B has independent columns (without
using the fact that A and B are both invertible, i.e., the result
of (a)).