Interesting problem on Eigenvector

Question

I face this problem in my exam I know how to find eigenvalue and eigenvector. But in this question I don't get any idea how to find eigenvector

Let the eigenvalue of a $2×2$ matrix $A$ be $1,-2$ with eigenvector $x_1$ and $x_2$ respectively.

Then the eigenvalue and eigenvector of the matrix $A^2-3A+4I$ ?

In this problem I easily find eigenvalue but on finding eigenvector I don't get idea. Please help.

Answer 1

$x_1$ and $x_2$ are still eigenvectors, as $$ (A^2-3A+4I)x_1=(1-3+4)x_1=2x_1,\quad(A^2-3A+4I)x_2=(4+6+4)x_2=14x_2. $$