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If A and B are n-rowed unitary matrices,...

If A and B are n-rowed unitary matrices,then AB and BA are also unitary matrices.

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A and B are two orthogonal matrices. Hence,
`A^(T)A=A A^(T)=I` and `B B^(T)=B^(T)B=I`
AB will be orthogonal if
`(AB)^(T) (AB)=(B^(T)A^(T)) (AB)`
`=B^(T) (A^(T)A)B`
`=B^(T) IB`
`=B^(T)B=I`
Similarly, we can show that `(BA)^(T) (BA)=I`. Therefore, Ba is also orthogonal.
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