Originally Posted by

**Thomas** Find all the eigenvalues (real and complex) of the matrix:

$\displaystyle \left( \begin{array}{ccc} -6 & 22 & 11 \\ -2 & -7 & -2 \\ -1 & 26 & 10 \end{array} \right)$

I started off by using the formula for finding the characteristic polynomial, and reducing the determinant matrix to:

$\displaystyle det\left( \begin{array}{ccc} 0 & 26x+134 & -x^2+4x+49 \\ 0 & x+59 & -2x+22 \\ 1 & -26 & x-10 \end{array} \right)$

Which then got me to:

$\displaystyle det\left( \begin{array}{ccc} 26x+134 & -x^2+4x+49 \\ x+59 & -2x+22 \end{array} \right)$

When I take the determinant of that, I get a cubic polynomial which is quite difficult to solve.

Is there an easier way to do this, or am I just doing it completely wrong? Can I possibly reduce it more?