# eigenvalue and eigenvector question

i got stuck in this question of my practice lab. Please give me some hint of how to achieve that question. thank you very much

'Suppose the matrix $A$ has eigenvectors $v$ with corresponding eigenvalues . Show that $v$ is an eigenvector of $A^n$. What is its corresponding eigenvalue?'

If $v$ is an eigenvector of $A$, then $$Av=\lambda v$$ Pre-multiply both sides by matrix $A$ $$A^2v=\lambda Av=\lambda^2v$$ Pre-multiply again by $A$ $$A^3v=\lambda^2Av=\lambda^3v$$ Can you see the pattern?