Diagonalisation and Kronecker Product

If $A$ is a $n\times n$ matrix with complex numbers for elements, and $C$ the $2\times2$ matrix defined by $$\begin{bmatrix} -2&4\\-3&5 \end{bmatrix}.$$ How do you prove that the Kronecker product $C\otimes A$ is diagonalisable if and only if $A$ is diagonalisable?

Use the properties of the Kronecker product: $$(P^{-1}\otimes Q^{-1})(C \otimes A) (P \otimes Q) = (P^{-1}\otimes Q^{-1})( CP\otimes AQ) = (P^{-1}CP)\otimes (Q^{-1}AQ)$$ The given matrix $C$ is diagonalizable (it has two different eigenvalues), thus, there exists $P$ such that $P^{-1}CP$ is diagonal.
If $A$ is diagonalizable, then clearly $C\otimes A$ is diagonalizable.
$C \otimes A$ is diagonalisable if and only if both $A$ and $C$ is diagonalisable.
• Given the fact that $C \otimes A$ is diagonalisable if and only if both $A$ and $C$ are diagonalisable and the fact that the given $C$ matrix is diagonalisable results in the fact that $C \otimes A$ is diagonalisable if and only if $A$ is diagonalisable.