We define $T(X) = AX$ with $A$ & $X$ square matrices of size $n$ with complex entries.
First, write down all the eigenvalues (with their respective algebraic multiplicities) of $T$. Use this to compute the trace & determinant of $A$.
I have proven that $A$ and $T$ have the same eigenvalues. My guess is that each eigenvalue of $A$ occurs with algebraic multiplicity $n$ times in order to give us a total sum of $n^2$, the dimension of the space. This is the part I'd like some help proving.
Possible idea : Compute $T$ for a basis which puts it into upper triangular form and then read off the diagonal entries for the eigenvalues.
Any help would be great.