Need some help with this, the book is written in a very confusing way. I'm following the examples exactly and not getting the correct result.

The matrix A = [2,-1 ; -1,2]

I found the two eigenvalues, to beλ= 1 andλ= 3

Now when I do the A -λI calculation and reduce the matrix I get

forλ = 1

x1 + x2 = 0 --> x1 = -x2

however, the solution in the book says the answer is

[1,1]

forλ = 3

-x1 + x2 = 0 --> x1 = x2

but again, the solution says

[-1, 1]

Can someone please clarify or point out what I am doing wrong?

Thanks.