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!

If $$M = \pmatrix{t & 1-t\cr 1-t & t}$$
show by induction that 
$$ M^j = \pmatrix{j t - (j-1) & j - j t\cr
j - jt & jt - (j-1)\cr} $$
and then take $j=n$.
