TITLE: How do you quickly find the eigenvalues of this matrix?
QUESTION [5 upvotes]: I have a final exam tomorrow, am sure a 3x3 eigen value problem like the one below is there.
But I find it very hard to find eigen values without zeros in the matrix
Show me how you do it quickly so that I can apply it tomorrow;
thanks

REPLY [12 votes]: You could (specially if you have a calculator at hand to crunch numbers) do this:

Get the trace (or sum of the diagonal elements). It is $32-25-2=5$
Compute the determinant. It is $-8$
Compute the square of the matrix and get the trace of it. It is $21$

You now know that:
$$\lambda_{1} \lambda_{2} \lambda_{3} = -8 \space\space\space\space\space(1)$$
$$\lambda_{1} + \lambda_{2} + \lambda_{3} = 5 \space\space\space\space\space(2)$$
$$\lambda_{1}^{2} + \lambda_{2}^{2} + \lambda_{3}^{2} = 21 \space\space\space\space\space(3)$$
Where $\lambda_{1}, \lambda_{2}, \lambda_{3}$ are the eigenvalues to work out. Now, let's say you feel lucky and want to assume that all the eigenvalues are integer. Then, from equation $(3)$ you know the largest one could be only $3$ or $4$ in absolute value, in which case the second largest would have to be $\pm 2$ and then $\pm 1$. If the largest were $3$, the second would have to be $3$ and the third couldn't be an integer. Now, from $(1)$ you know either one is negative or all are. Now, at once you see from $(2)$ that $4$ got to be positive, and the last choice of whether $1$ or $2$ is negative is solved by mentally trying, to find that the answer is indeed $-1$, $2$, $4$.
Yes, we assumed the answer to be made of integers, but, from experience, this is a good assumption to make to give it a first go in exams questions, as it's usually the case!