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Is it true for complex matrices $X,Y$ that $$ (XY)^*=X^* Y^*? $$ where $^*$ refers to complex conjugation. How can we prove this if so? Thanks!

Note: I am referring to complex conjugation, not the hermitan transpose. The answer below refers to hermitan tranpose.

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    Matrix multiplication is defined only using product and sum of entries, and the conjugation preserves both operations for complex field. – 2017-01-19
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    While your question is legitimate, please note that in linear algebra, the standard notation for the complex conjugate of a matrix $A$ is $\overline{A}$. The asterisk notation in almost all cases refers to the conjugate transpose, not complex conjugate. – 2017-01-20

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For conjugate transpose it holds $(XY)^*=Y^*X^*$, see here.

Edit: For complex conjugation, indeed, $(XY)^*=X^*Y^*$, see here, and the wikipedia article here, section "generalisations".

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    Op says $*$ denotes the complex conjugation, not hermitian. – 2017-01-19
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    I am talking about complex conjugation not the hermitan transpose. Thanks! – 2017-01-19
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    Yes, sorry for the confusion. Both use $*$. – 2017-01-19
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    @Integrals The entries of $XY$ are sums of products, where conjugation is the same as first conjugating and then form the sum of products. – 2017-01-19
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    @DietrichBurde Thanks, that makes sense. – 2017-01-19