How to find the eigenvalue? I have been given a matrix 
$$A = \begin{bmatrix}
1&1&0&0\\
1&1&0&0\\
0&0&1&1\\
0&0&1&1
\end{bmatrix}$$
I expanded by row one twice to get the characteristic polynomial: 
$(1-\lambda)^2[(1-\lambda)^2 -1] - 1[(1-\lambda)^2 - 1]$
Which I solved lambda and got that $\lambda = 0$. I checked my work with the answer in the back and its wrong. They got another value for lambda which I can't seem to get. 
http://i1317.photobucket.com/albums/t638/ayoshnav/Snapshot_20140326_zps80f4ba36.jpg
http://i1317.photobucket.com/albums/t638/ayoshnav/Snapshot_20140326_1_zpsf7f28df4.jpg

The characteristic polynomial you calculated is correct, however, it simplifies to $$p(\lambda) = \lambda^2(\lambda - 2)^2$$
There is more than one solution to the equation $p(\lambda)=0$.
In your notes, the mistake you made was at the very end. You got that $$(1-\lambda)^2 = 1$$ and from that, you concluded that $$(1-\lambda)=1.$$
Why is this wrong? Well, for example, $(-2)^2 = 4 = 2^2$, but that does not mean that $-2=2$, now does it?
