eigen values and vectors in this question

Question

I have the following matrix to be solved for landa. here is the matrix with my answer (sorry for the wrong equation writing format):

but the correct equation from this matrix should be (when simplified):

Please explain what I'm doing wrong

Answer 1

Laplace development with respect to the first row: \begin{align} \det\begin{bmatrix} 1-\lambda & 1 & 0 \\ 1 & -\lambda & -1 \\ -1 & 0 & 1-\lambda \end{bmatrix} &= (1-\lambda)\det\begin{bmatrix} -\lambda & -1 \\ 0 & 1-\lambda \end{bmatrix} - \det\begin{bmatrix} 1 & -1 \\ -1 & 1-\lambda \end{bmatrix} \\ &=-\lambda(1-\lambda)^2-(1-\lambda-1)\\ &=-\lambda(1-\lambda)^2+\lambda\\ &=\lambda(1-1+2\lambda-\lambda^2)\\ &=\lambda^2(2-\lambda) \end{align}

Answer 2

Your centre matrix component should be $-\lambda $ not $\lambda $

Then you would get:

$-\lambda (1-\lambda)^2-(1-\lambda-1)=-\lambda (\lambda^2-2\lambda)=0$

Thus $\lambda^2(2-\lambda)=0$

Answer 3

There is a sign error with $a_{22}$ as it should be negative.

We have:

$$(1-\lambda)(-\lambda(1-\lambda))-(1(1-\lambda)-1) = -\lambda^3 + 2 \lambda^2 = \lambda^2(2-\lambda) = 0$$