Question about free variables?

[chuy52506]

Homework Statement

Show that A=
|1 1 1 1|
|0 1 0 0|
|0 0 1 0|
|0 0 0 1|
is not diagonalizable.

Homework Equations

The Attempt at a Solution

I know i have to show the algebraic multiplicity does not equal the geometric mult. of the eigenvalue. So i found an eigenvalue to be 1 which then reduces the matrix to be:
|0 1 1 1|
|0 0 0 0|
|0 0 0 0|
|0 0 0 0|

And now i know that x₂=-x₃-x₄ with the free variables being x₁,x₃,x₄ but i do not know how to write this as a solution. Would it be
x₁[1 0 0 0]^T + x₃[-1 0 1 0]^T + x₄[-1 0 0 1]^T??

 

[lanedance]

so what is the algebraic multiplicity guessing it is 3

try testing the vectors you found on the original matrix, it should be clear x3 & x4 do not have eigenvalue of 1, in fact they do not satisfy the condition you stated, which is correct