$C=A+B$ diagonalizable $\implies$ $A$ and $B$ simultaneously diagonalizable?

Question

If $A$ and $B$ are simultaneously diagonalizable matrices, then clearly $A+B$ is diagonalizable. But is the converse true ? i.e if $C=A+B$ is diagonalizable, are $A$ and $B$ simultaneously diagonalizable?

Answer 1

Take any non-diagonalizable matrix $A$ and let $B=-A$. Then $A+B$ is the null matrix, which is diagonalizable.

Answer 2

$A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $B=\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$are not diagonalizable but $A+B$ is diagonal.