Is the product of any two invertible diagonalizable matrices diagonalizable?
A counterexample:$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}1&1\\0&-1\end{pmatrix}=\begin{pmatrix}1&1\\0&1\end{pmatrix}$$ The result is a well known nondiagonalizable matrix, the left matrix of the product is diagonal already, and the right matrix can be written as $$\begin{pmatrix}1&1\\0&-1\end{pmatrix}=\begin{pmatrix}-1&1\\2&0\end{pmatrix}\begin{pmatrix}-1&0\\0&1\end{pmatrix}\begin{pmatrix}-1&1\\2&0\end{pmatrix}^{-1}.$$