linear algebra - Finding eigenvalues of a matrix given its characteristic polynomial and the trace and determinant

I am told a matrix A has characteristic polynomial: $(\lambda−1)^3(a\lambda+\lambda^2+b),$ and that $\text {tr}(A)=12,$ and $\det(A) =14.$

I am asked to find the eigenvalues.

Is the only to do this by simply using the fact that the sum of the eigenvalues is the trace, and the product is the determinant?

I had done so, and had gotten that the eigenvalues are 1, 2 and 7, (1 with multiplicity 3), which was correct.

I am asking because won't the number of eigenvalues have to be less than the number of rows and columns of the matrix? What if, if given different trace and determinant values, I somehow found eigenvalues which satisfied the trace and determinant given, but had a number of eigenvalues that exceeded the number of columns and rows of the matrix?

Also, I was not given the dimensions of the matrix A.

Answer

I hope I understand everything you said.

1. The degree of your characteristic polynomial is 5. Therefore, the matrix it is coming from is $5 \times 5$.


2. An $n \times n$ matrix has $n$ (complex) eigenvalues (counting multiplicities). That's called the fundamental theorem of algebra.


3. Should the matrix be real and should you be only interested in real eigenvalues, you would see that some of the eigenvalues are conjugate from one another.


4. Given that the eigenvalues are the zeros of the characteristic polynomial, and given that it is factored, you directly notice that $\lambda = 1$ is an eigenvalue with algebraic multiplicity $3$. Therefore, you only have $5-3 = 2$ remaining eigenvalues to find.


5. You have two conditions (trace and determinant), i.e. two equations, and solve for two eigenvalues. This shouldn't be an issue and solvable for (almost) every pair (trace,det).