Powers of small square matrices over the Laurent polynomial ring with integer coefficients I'm trying to calculate the powers of a 2 by 2 matrix with entries in $\mathbb{Z} \left[ t,t^{-1}  \right]$.
The matrix is \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix}
I tought of writing my matrix in the following form 
$$ \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix} = t \begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix} + \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix} $$ 
Now we get 
$$ \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix} ^n = \sum_{k=0}^n \binom{n}{k} t^{n-k} \begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix}^{n-k} \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}^{k} $$
The only issue is that if $\mathcal{M}_n(\mathbb{Z} \left[ t,t^{-1}  \right])$ is not isomorphic to $\mathcal{M}_n(\mathbb{Z}) \left[ t,t^{-1}  \right] $ then my logic will be wrong.
I also thought of diagonalizing the matrix, but I found no algorithme to get it done.
Thanks in advance for your help.
 A: This is not much different from Robert Israel's answer, but here goes:
Let $A(t)=\begin{pmatrix}0&1\\1&t\end{pmatrix}$ and write $A^n(t)=\begin{pmatrix}c_n&b_n\\b_n&a_n\end{pmatrix}$. Then, 
$$\begin{pmatrix}c_{n+1}&b_{n+1}\\ b_{n+1}&a_{n+1}\end{pmatrix}=A^{n+1}=\begin{pmatrix}c_n&b_n\\b_n&a_n\end{pmatrix}\begin{pmatrix}0&1\\1&t\end{pmatrix}
=\begin{pmatrix}b_n&c_n+tb_n\\a_n&b_n+ta_n\end{pmatrix}$$
implies $c_{n+1}=b_n, b_{n+1}=a_n, a_{n+1}=b_n+ta_n$. That means, $A^n=\begin{pmatrix}a_{n-2}&a_{n-1}\\a_{n-1}&a_n\end{pmatrix}$ and
$a_n=ta_{n-1}+a_{n-2}$ with initial conditions $a_{-1}:=0$ and $a_0=1$. This recurrence can be utilized to prove the following explicit formula
$$a_n(t)=\sum_{k=0}^{\lfloor\frac{n}2\rfloor}\binom{n-k}kt^{n-2k}.$$
A: Your matrix (call it $A(t)$) has characteristic polynomial $\lambda^2 - t \lambda - 1$, so it satisfies $A(t)^2 - t A(t) - I = 0$ and thus
$A(t)^{n+2} = t A(t)^{n+1} + A(t)^n$.  For $n \ge 2$ I get 
$$ A^n = \pmatrix{i^{n-2} U_{n-2}(-it/2) & i^{n-1} U_{n-1}(-it/2)\cr
     i^{n-1} U_{n-1}(-it/2) & i^n U_{n}(-it/2)\cr} $$
where $U_n$ are the Chebyshev polynomials of the second kind. 
