Given the Matrix $A$ $\in$ $\mathbb {R}^{4\times4}$ with the following known properties:
$|Spec(A)| = 3$
$Tr(A) = 92$
$Tr(A^2) = 4060$
$Rank(A - E)= 2$
I need to determine the determinants of this matrix. So far I'm gathering that since $|Spec(A)|$ = 3 I have 3 eigenvalues and because the determinant is the product of all the eigenvalues I have $ det(A) =\lambda_1\lambda_2\lambda_3 $
From 4. I gather that $Rank(A-E) = 2 < 4 \iff \lambda = 1$ is an eigenvalue of $A$
The relationship between determinant and trace as well as the trace of $A^2$ is not clear to me aside from one of the coefficients of the characteristic of the equation being written in the form of the trace but not for an $n$ this high.
Thank you all in advance!