I've got the following question
Consider the linear operator of left multiplication by an $m \times m$ matrix $A$ on the vector space of all $m \times m$ matrices. Determine the trace and determinant of this operator.
I'm a bit stuck as to how to even begin, I know this is going to involve eigenvalues/vectors and that if $\lambda_1, \lambda_2, ... ,\lambda_m$ are the $m$ roots of the characteristic polynomial of an $m \times m$ matrix $A$, then:
$\det(A) = \lambda_1 ... \lambda_m$
and
$\text{trace}(A) = \lambda_1 + ... + \lambda_m$
But obviously not all $m \times m$ matrices have $m$ eigenvalues so I'm really stuck on this question.
Thanks!