Math Help - Normal matrix

1. Normal matrix

Show that if A is Mn(C) is normal then Ax=0 if and only if A*x=0
Thank you!
Does this hold in general? I am not sure.

2. Use the inner product: $Ax=0\ \Longleftrightarrow\ \langle Ax,Ax\rangle=0 \ \Longleftrightarrow\ \langle A^*Ax,x\rangle=0\ \Longleftrightarrow\ \ldots$ (now use the fact that A is normal).

The result is false in general, for example if $A = \begin{bmatrix}0&1\\0&0\end{bmatrix},\ x = \begin{bmatrix}1\\0\end{bmatrix}$.