# Finding eigenvalues and eigenvectors of a $3\times 3$ matrix

My matrix is $$A=\begin{pmatrix}0 & 0 & -2\\ 1 & 2 & 0 \\ 0 & -2 & 0\end{pmatrix}$$

I have to find its eigenvalues and eigenvectors but characteristic polynomial is $3$rd degree and I can't calculate it. Please give a help for it. Thanks.

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Do you mean that you can't factor the polynomial or that you can't find it? In either case, what have you tried? –  EuYu Nov 12 '12 at 6:47
i have found the characteristic polynomial as x^3-2x^2-4=0, solved it with calculator and find x1= 2.5943 and others are complex roots. How can i make calculations in this way? –  aicha Nov 12 '12 at 6:54
Well, the roots that you've found are your three eigenvalues. For each eigenvalue, row reduce $A-\lambda I$ as usual to find your eigenvectors. There's nothing different than the usual routine, except maybe that the numbers are not as nice as you're used to. –  EuYu Nov 12 '12 at 7:02
as i understood,there is no way without making these calculations.i thought maybe there could be a different way to find eigenvectors. ok. thank you. –  aicha Nov 12 '12 at 7:12

The eigenvalues of a matrix A are the solutions $\lambda$ to the equation of the form $det(A-\lambda I)$. Det refers to the determinant of the matrix formed by $(A - \lambda I)$ and I is the $n$ x $n$ identity matrix -- this is called the characteristic equation. To find your eigenvalues and eigenvectors just find the solution using $det(A-\lambda I)$. If you need me to elaborate any further just ask.