Find all the eigenvalues (real and complex) of the matrix:
I started off by using the formula for finding the characteristic polynomial, and reducing the determinant matrix to:
Which then got me to:
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?
When you see that, you know that you have struck lucky, because there is a factor 3+x going down the middle column. Take this factor out, and you are left with . Now add twice the middle row to the bottom row, and expand down the middle column, to get . There's the eigenvalue equation, neatly factorised, without having to do any heavy calculations at all!