Multiplicity of Eigenvalues in a 3x3 Matrix

[icefall5]

Homework Statement

The matrix $A = \begin {bmatrix} 0 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{bmatrix}$ has two real eigenvalues, one of multiplicity 2 and one of multiplicity 1. Find the eigenvalues and a basis of each eigenspace.

Homework Equations

N/A

The Attempt at a Solution

I've done cofactor expansion to come up with the equation $- \lambda^3 - \lambda^2$, and that the eigenvalues are therefore -1 and 0, but I don't know how to determine the multiplicity of each. I've looked it up and have gotten nowhere. I can do the rest of the problem, I just don't know how to get these multiplicities.

EDIT: Working on the problem further, and I got the basis for -1. There are supposed to be two vectors for 0, however, but the RREF of the matrix is $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$. How does that translate to two eigenvectors?

Thanks in advance for any help!

 

[Mark44]

Homework Statement

The matrix $A = \begin {bmatrix} 0 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{bmatrix}$ has two real eigenvalues, one of multiplicity 2 and one of multiplicity 1. Find the eigenvalues and a basis of each eigenspace.

Homework Equations

N/A

The Attempt at a Solution

I've done cofactor expansion to come up with the equation $- \lambda^3 - \lambda^2$,

That's not an equation. The equation is $- \lambda^3 - \lambda^2$ = 0, or $\lambda^2(\lambda + 1) = 0$
Can you see that one of the eigenvalues has multiplicity 2 and the other has multiplicity 1?

and that the eigenvalues are therefore -1 and 0, but I don't know how to determine the multiplicity of each. I've looked it up and have gotten nowhere. I can do the rest of the problem, I just don't know how to get these multiplicities.

EDIT: Working on the problem further, and I got the basis for -1. There are supposed to be two vectors for 0, however, but the RREF of the matrix is $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$. How does that translate to two eigenvectors?

Thanks in advance for any help!

 

[icefall5]

Ah, yes, I just couldn't simplify that properly; sorry! Thank you!

[STRIKE]But now the second part -- how does that matrix equate to two eigenvectors?[/STRIKE]

I'm an idiot; I figured it out. Thanks again!