Math Help - finding eigenvectors

1. finding eigenvectors

Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $
$A = \begin{array}{ccc} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \\ \end{array}$$

I got the eigenvalues $\lambda_1=\lambda_2=\lambda_3=0$
To find the associated eigenvectors $(0I_3-A)\vec x = \vec 0$
This gives me the matrix $
\begin{array}{cccc}
0 & -1 & -2 & 0 \\
0 & 0 & -3 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}$
where the last column of 0s is the augmented part of the matrix.
This matrix is row equivalent to the identity matrix and that confuses me

2. Originally Posted by Jskid
Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $
$A = \begin{array}{ccc} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \\ \end{array}$$

I got the eigenvalues $\lambda_1=\lambda_2=\lambda_3=0$
To find the associated eigenvectors $(0I_3-A)\vec x = \vec 0$
This gives me the matrix $
\begin{array}{cccc}
0 & -1 & -2 & 0 \\
0 & 0 & -3 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}$
where the last column of 0s is the augmented part of the matrix.
This matrix is row equivalent to the identity matrix and that confuses me
$\displaystyle\begin{bmatrix}0&1&2\\0&0&3\\0&0&0\en d{bmatrix}\Rightarrow\text{rref}=\begin{bmatrix}0& 1&0\\0&0&1\\0&0&0\end{bmatrix}$

$\displaystyle\left\{\begin{bmatrix}1\\0\\0\end{bma trix}\right\}$