Solution
= Solution
This is a <generalized eigenvalue problem>. A nonzero <vector> lies in the <kernel> of $A-\lambda B$ precisely when its <determinant> vanishes:
$$
0=\det\begin{pmatrix}1&2-2\lambda\\2-2\lambda&1\end{pmatrix}
=1-4(1-\lambda)^2.
$$
Hence
$$
\boxed{\lambda=\frac12\quad\text{or}\quad\lambda=\frac32}.
$$
The corresponding <eigenvectors> can be taken as $(1,-1)^T$ and $(1,1)^T$, respectively, which also checks that both values give nontrivial solutions.