Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$ Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. 
If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be determinant and trace of matrix $A+B$?
 A: Since $A$ is $4\times 4$ and its eigenvalues are $2$, $-2$, $1$, and $-1$, the minimal and characteristic polynomials of $A$ agree and are both equal to
$$(t-1)(t+1)(t-2)(t+2) = (t^2-1)(t^2-4) = t^4 - 5t^2 + 4.$$
In particular, by the Cayley-Hamilton Theorem,
$$A^4 - 5A^2 + 4I = 0,$$
and therefore
$$B+A = A^4 - 5A^2 + 5I + A = (A^4-5A^2+4I) + (A+I) = A+I.$$
Now notice that $\lambda$ is an eigenvalue of $A$ if and only if $\alpha\lambda+\beta$ is an eigenvalue of $\alpha A+\beta I$, to conclude that the eigenvalues of $B+A=A+I$ are $0$, $-1$, $2$, and $3$. Therefore, the trace is $0-1+2+3 = 4$, and the determinant is $0$ (since $A+I$ is not invertible, or since the determinant is the product of the eigenvalues). 
A: Pick one of the eigenvectors of $A$, call it $\bf v$, with corresponding eigenvalue $\lambda$. Can you work out $(A+B){\bf v}$? From that, can you work out the determinant and trace?
A: Forgive me if I mistake what Gerry is trying to say, but I believe it is this:


*

*If $v$ is an eigenvector of $A$, then $v$ is an eigenvector of $A+B$ (what is the associated eigenvalue, then?). Use the formula for $B$, and note that $A^n(v) = \lambda^nv$.

*From the associated eigenvalues you get for $A+B$, you should be able to explicitly state the characteristic polynomial for $A+B$.

*For a 4x4 matrix, the trace is the negative of the coefficient of the cubic term in the characteristic polynomial, and the determinant is the constant term.
A: Start by finding the eigenvalues of $B$, using the eigenvalues of $A$. Find the eigenvalues of $A+B$ using this.
Then recall that the determinant here will be the product of the eigenvalues of $A+B$ and the trace will be the negative of the sum of the eigenvalues of $A+B$.
