Hermitian Matrix true or false?

Question

I am trying to prove or disprove the statement:

If $A,B$ are n by n complex matrix, then if $A+B$ is Hermitian, then both $A$ and $B$ are Hermitian.

Is this statement true or false?

Answer 1

A real matrix is Hermitian if it is symmetric. Use

$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0\\ 1 & 0 \end{bmatrix}+\begin{bmatrix} 0 & 0\\ -1 & 1 \end{bmatrix}$

LHS is hermitian but the matrices in RHS are not Hermitian.

Answer 2

$$\begin{bmatrix}0&i\\-i&0\end{bmatrix}=\begin{bmatrix}0&0\\-i&0\end{bmatrix}+\begin{bmatrix}0&i\\0&0\end{bmatrix}$$

Answer 3

Let $A$ be a matrix, which is not hermitian and $B=-A$, then $A+B$ is hermitian.