Let A and B be square matrices of the same size. Prove that if A and B are both upper triangular, then AB is upper triangular. If A and B are both upper triangular, do they necessarily commute? If you think they must commute, give a proof; if you think they might not commute, give a counterexample.

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Let A and B be square matrices of the same size.

Prove that if A and B are both upper triangular, then AB is upper triangular.

If A and B are both upper triangular, do they necessarily commute? If you think they must commute, give a proof; if you think they might not commute, give a counterexample.

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