I am having some confusion over how I would attack a proof of the properties of matrix adjoints. Here is an example:
Let $A,B\in M_{n\times n}(\mathbb{F})$ with adjoints $A^*$ and $B^*$. Prove $(A+B)^*=A^*+B^*$.
Now I know what I would do for the similar problem:
Let $S\colon V\to V$ and $T\colon V\to V$ be linear operators, and $V$ has inner product $\langle\cdot,\cdot\rangle$. Prove $(S+T)^*=S^*+T^*$.
This would be done by:
\begin{align} \langle(S+T)^*(x),y\rangle& =\langle x,(S+T)(y)\rangle \\ & =\langle x,S(y)+T(y)\rangle \\ & =\langle x,S(y)\rangle+\langle x,T(y)\rangle \\ & =\langle S^*(x),y\rangle+\langle T^*(x),y\rangle \\ & =\langle S^*(x)+T^*(x),y\rangle \\ & =\langle(S^*+T^*)(x),y\rangle. \end{align}
I am confused though, as to how the first proof (with the matrix adjoint) would be done differently than when it is done with the linear operator adjoint.
It seems that for matrices the easiest procedure is to write down the $(i,j)$'th entry of $(A+B)^*$ and convince oneself that it is equal to the $(i,j$)'th entry of $A^*+B^*$.