Given a $2 \times 2$ matrix $B$ that satisfies $B^2=3B-2I$, find the eigenvalues of $B$.
Let $v$ be an eigenvector for B, and $\lambda$ it's corresponding eigenvalue. Also, let $T$ be the linear transformation (not that this is exactly necessary for the question, but just added it in for my understanding.) Therefore,
$$T(v) = Bv = \lambda v$$
Now I'm unsure how to incorporate this information into the quadratic equation given above since by matrix / vector arithmetic isn't extremely solid. Thanks!