Prove or disprove: If $A^2$ is normal matrix then $A$ is normal matrix 
Prove or disprove: If $A^2$ is normal matrix then $A$ is normal matrix. 

I think this is wrong but simply can't build a counterexample.
Any hints on how to build a counterexample? There are many conditions and it's hard for me to find a matrix.
Note that: $A$ is normal if $AA^* = A^*A$.
 A: Pick 
$$ A = \begin{pmatrix} u & v \\w & -u\end{pmatrix}$$
Now 
$$A^2 = (u^2+v\,w) \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}$$
is clearly normal.
But for arbitrary $u$, $v$, $w$, $A$ would not be normal.
Just pick $w \neq v$
A: Take the matrix
$$A = \begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$
Then we have $A^2 = 0$, which is normal. On the other hand,
$$AA^* = \begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}=\begin{pmatrix}1&0\\0&0\end{pmatrix}$$
$$A^*A=\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}=\begin{pmatrix}0&0\\0&1\end{pmatrix}$$
so that $A$ itself is not normal.
A: This is to answer OP's question in the comment section. The method I outline is a general way to attack problems of this nature.
Clearly there are no counter examples in $1\times 1$ matrices.
Try $2 \times 2$ matrices. 
Let
$$D(A) = A^*A - AA^*$$
and let us work in the field of reals.
Let $A$ be
$$
A = \begin{pmatrix}u & v\\w& x\end{pmatrix}$$
Then 
$$D(A) = \begin{pmatrix}\left( w-v\right) \,\left( w+v\right)  & \left( w-v\right) \,\left( x-u\right) \cr \left( w-v\right) \,\left( x-u\right)  & -\left( w-v\right) \,\left( w+v\right) \end{pmatrix}$$
and
$$
D(A^2) = \begin{pmatrix}\left( w-v\right) \,\left( w+v\right) \,{\left( x+u\right) }^{2} & \left( w-v\right) \,\left( x-u\right) \,{\left( x+u\right) }^{2}\cr \left( w-v\right) \,\left( x-u\right) \,{\left( x+u\right) }^{2} & -\left( w-v\right) \,\left( w+v\right) \,{\left( x+u\right) }^{2}\end{pmatrix}$$
If you let $x = -u$ and $w \neq v$ then you get $D(A) \neq 0$ and $D(A^2) = 0$.
Note that $D(A)$ term-by-term divides $D(A^2)$. So at least for $2\times2$ matrices
$D(A)=0 \Rightarrow D(A^2)=0$
