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?
Just for future reference, these questions are often rigged so that you can take factors out of the determinant (by doing elementary row and column operations) without having to multiply it out. In this case, the 22 and the 11 in the top row of the matrix suggest that you might try subtracting twice the right column from the middle column. Then you get
.
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!