I have a question regarding a 3x3 matrix and its eigenvalues. The matrix is $A= \begin{bmatrix}1 & 1 & 2\\-2 & -1 & 2 \\ -1 & -2 & 3\end{bmatrix}$
What I have attempted is to let $det(A-\lambda I)=0$, which resulted in the cubic, $-\lambda ^3+3\lambda ^2 -7\lambda +11 =0$
I believe I have worked out the correct polynomial, but without the help of a computer or the equation for cubic roots I am unsure on how to solve this equation. Normally my method would involve guessing a solution (a factor of $\lambda^0$, in this case 11) and then doing polynomial division. In this case the polynomial has one real root and a complex conjugate.