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Exercise Let A and B be square matrices. Prove that A and B are invertible ⇔ AB is invertible. Please also explain why the square condition is necessary.

Answer In the lecture, we have already shown the direction ⇒.

Now we assume AB is invertible and try to prove A and B are invertible. Let C be the inverse of AB. Then we have ABC = I and CAB = I. By this criterion, the fact that A(BC) = I and A is square implies A is invertible. Similarly, the fact that (CA)B = I and B is square implies B is invertible. This completes the proof for the direction ⇐.

The square condition is necessary because nonsquare matrices are not invertible. For a concrete example, consider

A = [ 1 -2 ], B = [ 3 ]
1

We have AB = I, but A and B are not invertible.