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

linear-algebra matrices diagonalization

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


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

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