Eigenvector Q, working out the eigenvalues

[Sonal7]

sound like an easy question but I cant make head or tails of the answer. Heres my working to the question. But the answer book has a 3D matrix from a 2D matrix.
My answer looks incorrect. Why isnt my method working? The full solutions page wouldnt fit in the screen shot as its over two different pages but give clues to the correct method. I wonder what their approach is? Do i need to go back and take a look at textbooks?

 

[Sonal7]

I am 100% sure this one is a mistake on the part of the publishers. I dont think the answer matches the Q. I feel bad I didnt spot it. I wish to delete it as it wrong. I spend a lot of time skipping over answers from intergral as there are tons of mistakes. So bad for your confidence.

 

[Romsek]

the answer is obviously incorrect.

 

[Sonal7]

I m so fed up of this. It so knocked my confidence.

 

[Romsek]

I m so fed up of this. It so knocked my confidence.

I'd like to welcome you to planet Earth, and wish you an enjoyable stay.

That said your workings for eigenvalue 1 are incorrect. (really sloppy mistake probably from being flustered)