What is the geometric multiplicity of \lambda=0 as an eigenvalue of A?

[chuy52506]

Homework Statement

$$\lambda$$=0 is an eigenvalue of
A=
|1 1 1 1 1|
|1 1 1 1 1|
|1 1 1 1 1|
|1 1 1 1 1|
|1 1 1 1 1|

Homework Equations

Find the geometric multiplicity of $$\lambda$$=0 as an eigenvalue of A

The Attempt at a Solution

I row reduced it then got the last four rows of all 0s but don't know where to go from there??

 

[chuy52506]

now i have x₁=-x₂-x₃-x₄-x₅
does that mean it has geom mult of 5?

 

[spamiam]

Now you have 4 free variables, namely $x_2, x_3, x_4, x_5$. If that doesn't tell you right away, set $x_2 = t_1$, $x_3 = t_2$, etc. This gives you
$$\begin{pmatrix}<br /> x_1 \\<br /> x_2 \\<br /> x_3 \\<br /> x_4 \\<br /> x_5<br /> \end{pmatrix}=<br /> \begin{pmatrix}<br /> -t_1 - t_2 - t_3 - t_4 \\<br /> t_1 \\<br /> t_2 \\<br /> t_3 \\<br /> t_4<br /> \end{pmatrix}$$

Now separate out the different variables and see how many basis vectors this gives you.