Given eigenvalue and eigenvector Find $A^{101} $

Question

I need help solving this problem Assume $\lambda =-1$ is an eigenvalue of a 3x3 matrix A and $x= [2\ 3\ 4]^{T}$ is an eigenvector corresponding to this $\lambda$. Find $A^{101}x$. I have no idea how to use the eigenvalue and vector to work my way up to $A$. I do know that $A^{101}=PD^{101}P^{-1}$ where $P$ is a matrix made from the eigenvectors as its columns and $D$ is a diagonal matrix

Answer 1

Hint:

For eigenvalues $\lambda$, you would have $Ax = \lambda x$, so you would have $A^2x=A\lambda x=\lambda^2x$. Then you can go on from this logic.

Answer 2

Hint: If $Ax = \lambda x$, then $$A^2x = A\lambda x = \lambda Ax = \lambda^2 x$$ Can you take it from here?