Let $P$ be a $2 \times 2$ complex matrix such that Trace(P) = 1, Determinant(P) = -6, Find Trace$(P^4 - P^3).$
What I have done, if $\lambda_1, \lambda_2$ is the eigen value, then I find $\lambda_1 = 3$ and $\lambda_2 = -2.$ So the eigen value of $P^4$ will be 81 and 16. From this point we can find Trace$(P^4).$ Similarly for Trace$(P^3).$ So we can find the desired result.
My question is am I right? Also What is the role of complex matrix here?