SOLUTION: I have worked out Eigenvalues of the following matrix 1 -1 2 0 2 0 3 -3 2 They are = -1, 2, 4 I cant seem to work out the eigenvector for -1. Could someone show me step

Algebra ->  Matrices-and-determiminant -> SOLUTION: I have worked out Eigenvalues of the following matrix 1 -1 2 0 2 0 3 -3 2 They are = -1, 2, 4 I cant seem to work out the eigenvector for -1. Could someone show me step       Log On


   

Question 361049: I have worked out Eigenvalues of the following matrix
1 -1 2
0 2 0
3 -3 2
They are = -1, 2, 4
I cant seem to work out the eigenvector for -1. Could someone show me step by step how this achieved. I can then use this to work out 2, and 4.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
For λ = -1, replacing -1 into the homogeneous system A - λI = O, we get the augmented homogeneous system
%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C3%2C0%2C0%2C3%2C-3%2C3%2C0%29%29.
~%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C1%2C0%2C0%2C1%2C-1%2C1%2C0%29%29Divide r2 and r3 by 3.
~%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29Add row2 to r3
~%28matrix%283%2C4%2C+0%2C-1%2C0%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29-2*r3 + r1
~%28matrix%283%2C4%2C+0%2C0%2C0%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29Add r2 to r1
~%28matrix%283%2C4%2C+1%2C0%2C1%2C0%2C0%2C1%2C0%2C0%2C0%2C0%2C0%2C0%29%29Interchange r1 and r3.
Thus y = 0 and x + z = 0, or z = -x. hence,
. The basis for eigenspace for λ = -1 is then +%28matrix%283%2C1%2C1%2C0%2C-1%29%29+, and this is its eigenvector.