Deducing the matrix $U^n$.

Question

Let $u$ be an endomorphism of $\mathbb{R^2}$ with the matrix in the standard basis $B$ is :U = $\begin{bmatrix} 5&1\\-4&0\end{bmatrix}$.

1.Give the matrix $U'$ of $u$ in the basis B' = {$v1=(1,-4) , v2 = (1,-1)$}.

2.What is the transition matrix $P$ from $B$ to $B'$. Calculate $P^{-1}$

3.Deduce the matrix $U^n$ for all integer $n>2$

I've solved the first two questions and found:

$U'$=$\begin{bmatrix} 1&0\\0&4\end{bmatrix}$ , $P$=$\begin{bmatrix} 1&1\\-4&-1\end{bmatrix}$ , $P^{-1}$=$\frac{1}{3}\begin{bmatrix} -1&-1\\4&1\end{bmatrix}$

But i have no idea on how to start solving the third question.

Help and hints are appreciated!

Thanks in advance.

Answer 1

What you have found is that $U=P^{-1}U'P$, so that $U^n=P^{-1}U'^nP$, as the $P$s and $P^{-1}$ cancel. $U^n$ should be easy to compute.