Math Help - Eigenvalue Method

Apply the eigenvalue method to find a general solution to:

x1' = x1 + 2x2 + 2x3
x2' = 2x1 + 7x2 + x3
x3' = 2x1 + x2 + 7x3

Here's what Ive done:

The matrix form of the system is:

x' = 1 2 2
.......2 7 1
.......2 1 7

Then the book says to do this:

A - &I = 1-&....2......2
..............2....7-&....1
..............2.....1.....7-&

I can't follow the rest. Can someone show how to do the rest? Thanks

Next thing to do is solve



Then follow the rest of this example

Pauls Online Notes : Differential Equations - Real Eigenvalues

Ok I get 3 eigenvalues: 0,6,9

Case 1:
I plug 0 into the matrix from the original post and notcie that its a singular matrix b/c the det is 0. In a 2x2 matrix, this means I can arbitrary choose a and solve for b. How does this work for my 3x3 matrix? Thanks

Well, what do you get for your row reduction? Incidentally, you can type up matrices as follows:

and augmented matrices this way:



Just double-click on the matrices to see how I entered them, and then enclose that code in math tags.