This is my attempt so far: Suppose I have two unitary matrices, A and B, such that $A^*A=I$ and $B^*B=I$. We want to show that $(AB)^*(AB)=I$. So we have that $$(AB)^*(AB)=B^*A^*AB=B^*IB=I.$$ Does this work as a proof? Do I need to show anything else?
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1Looks good ${}$ – user251257 Oct 04 '16 at 23:31
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1You have it! Well done. – Mark Viola Oct 04 '16 at 23:33