Linear Algebra Help: Calculating Eigenvalues & Eigenvectors of Matrix A

[GregoryGr]

Homework Statement

Calculate the eigenvalues and eigenvectors of the matrix:
$$ A= \begin{bmatrix}
3 & 2 & 2 &-4 \\
2 & 3 & 2 &-1 \\
1 & 1 & 2 &-1 \\
2 & 2 & 2 &-1
\end{bmatrix} $$

Homework Equations

nothing

The Attempt at a Solution

I've found the eigenvalues, but what disturbes me, is that I can't find a way to make the determinant triangular, as to find the values faster. Can anybody see a way to do that?

 

[Simon Bridge]

You wouldn't make the determinant triangular, the determinant is just one number.

You can make the matrix triangular by row-reduction:
- number the rows top to bottom 1-4.
- reorder the rows: 3-2-4-1 --> 1-2-3-4
- after that the row-reduction to upper-triangular form should come easily.

You probably want to do this for each eigenvalue to find the eigenvectors - so the best order for the rows will be different each time.

You want to try this for the eigenvectors - consider:
http://www.millersville.edu/~bikenaga/linear-algebra/eigenvalue/eigenvalue.html