Suppose that A is an m n matrix and B is an n p matrix.

Question:

Suppose that A is an m × n matrix and B is an n × p matrix. Prove that the column space of AB is a subset of the column space of A, that is C(AB) ⊆ C(A). Provide an example where the opposite is false, in other words give an example where C(A) ⊈ C(AB).