If A has r independent columns and B has r independent rows, AB is invertible. Proof
Chapter 4, Problem 34(choose chapter or problem)
If A has r independent columns and B has r independent rows, AB is invertible. Proof: When A is m by r with independent columns, we know that AT A is invertible. If B is r by n with independent rows, show that BBT is invertible. (Take A = BT.) Now show that AB has rank r. Hint: Why does AT ABBT have rank r? That matrix multiplication by AT and BT cannot increase the rank of AB, by 3.6:2
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