sum of two matrices question given condition

Question

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?

Answer 1

If an $n\times n$ matrix $A$ is orthogonal diagonalizable $\Leftrightarrow$ $A$ is symmetric.

Then if $A$, $B$ are two $n\times n$ matrices and orthogonal diagonalizable then they are symmetric.

Hence $A+B$ is also orthogonal diagonalizable since $A+B$ is obviously symmetric.