Q:

Prove or Disprove: If $A$ and $ B$ are $n \times n$ orthogonally diagonalizable matrices, then $A + B$ is orthogonally diagonalizable.

So I know the properties that $A^T = A$ if its orthogonally diagonalizable and also $A=PDP^T$, But cant quite use them to prove or disprove. Would appreciate any help.

A:

If $A^T=A$ and $B^T=B$, then $(A+B)^T=A^T+B^T=A+B$.

