(Solved): Dopending upon the numbers you are given, the matrix in this problem might have a characteriste pol ...

Dopending upon the numbers you are given, the matrix in this problem might have a characteriste polynomial that is not feasible to factor by hand without using methods from precalculus such as the rational root test and polynomial division. On an exam, you are expected to be able to find eiganvalues using colactor expansions for matrices of size \( 3 \times 3 \) or larger, but we will not expect you to go the extra step of applying the rational root test or performing polynomial division on Math 1553 exams. With this in mind, it you are unable to factor the characteristio polynomial in this particular problem, you may use a calculator or computer algobra system to get the eigervalues. The matro \[ A=\left[\begin{array}{cccc} -2 & 0 & 0 & 0 \\ 0 & -2 & 0 & 2 \\ -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \] has two real eigenvalues \( \lambda_{1}<\lambda_{2} \). Find these eigenvalues, their mutiplicities, and the dimensions of their corresponding oigonspaces. The emaller eigenvalue \( \lambda_{1}= \) has algobraic muliplicity and the dirmension of ts corresponding eigenspsece is The larger eigenvalue \( \lambda_{2}= \) has algobraic multplicty and the dimension of its corresponding eigenspace is Do the dimentions of the eigenspaces for \( A \) add up to the number of columns of \( A \) ?