Unitary matrix matrix problem I need to proof this question Show that for any
Recall that the unitary group $U(n)$ consists of all $A \in M_n(C)$ with $A^*A = I$. Show that a matrix $A \in M_n(C)$ is in $U(n)$ if and only if $\langle Ax,Ayi\rangle = \langle x,y\rangle$ for all $x, y \in C$.
So I just need the $\Leftarrow$ implication; I made it but I am not sure if my justification is good enough. 
$\langle Ax,Ay\rangle = \langle x,A^*Ay\rangle = \langle x,y\rangle = x^*A^*Ay = x^*y$ so {problem here} $A^*A = I$
therefore $A \in M_n(C)$ is in $U(n)$.
So do you guys think that this is good enough or do is their some problems in this argument?

Hints:
1.) For $T,S\in M_n(\Bbb C)$ we have $\ T=S\ \iff\ \forall x,y:\langle Tx,y\rangle=\langle Sx,y\rangle$. 
2.) Use the adjointness property: $\langle z,Ay\rangle=\langle A^*z,y\rangle$.
