A power of a matrix I am trying to find 
$$\begin{bmatrix}t&1-t\\1-t&t\\\end{bmatrix}^n$$ in $\mathbb{Z}_n[t^{\pm 1}]/(t-1)^2$, where $n$ is a positive integer. My guess is that the result is the identity matrix but am not sure about a proof. Thanks for any help or hints!

A variant of avs's strategy in comments: Write
$$ \begin{bmatrix} t & 1-t \\ 1-t & t \end{bmatrix}  =
I_{2\times 2} + (t-1)\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} $$
and use the binomial theorem (which applies because the $I$ commutes with everything) to raise this to the $n$th power.
All but two of the resulting terms contain two or more factors of $t-1$ and therefore vanish. One the the remaining terms has a factor of $\binom n1$ which equals $0$ modulo $n$, so it vanishes too. The last one is $\binom 11 I^n = I$.
