Find Complex Eigenvalues for 3x3 Matrix with All 9 Numbers at .3 | Homework Help

[maximade]

Homework Statement

A 3x3 matrix with all 9 of the numbers being .3
Find all the eigenvalues.

Homework Equations

The Attempt at a Solution

I worked through it and I ended up with (l=lamda) l^3-.9l^2+.54l-.162=0
With my calculator I found one of the values, which means that there are 2 complex values.
Considering how I have not done this in a while, how do I find the complex roots of this equation?

Thanks in advance

 

[vela]

You made a mistake calculating the characteristic polynomial. The three eigenvalues for that matrix are 0.9, 0, and 0.

To answer your question about the roots: in general when you have a cubic and one of its roots r, divide the cubic by (x-r) and find the roots of the resulting quadratic.

 

[HallsofIvy]

It should be obvious that one of the eigenvalues is 0 because this matrix is obviously not invertible. In fact, without doing any work at all it should be clear that the "row-echelon form" for this matrix is
$$\begin{bmatrix}3 & 3 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$$
so that all of R³ is mapped to R¹- the null space must be two-dimensional so 0 must be a "double" eigenvalue as vela says.