How to prove my matrix's powers? Being bored I played around with matrices I stumbled upon this matrix.
$$T = \left[\begin{array}{ll|cc}
0&2&0&0\\
\frac{1}{2}&0&0&0\\
\hline 1&0&1&0\\
1&1&0&1
\end{array}\right]$$
What is $T^{2^k}$ for $\cases{k\in \mathbb N\\k>1}$? I have my suspicions what it should be, but how can we prove it?

Well, let 
$$
T = \begin{pmatrix}A & 0 \\ B & I\end{pmatrix}, \quad A = \begin{pmatrix}0 & 2 \\ \frac{1}{2} & 0\end{pmatrix}, \quad B = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix},
$$
and $I$ is a unit matrix. One may see (by simple induction) that 
$$
T^n = \begin{pmatrix}A^n & 0 \\ B\sum_{i=0}^{n-1}A^i & I\end{pmatrix}.
$$
Also one may note that $A^2 = I$. Thus $A^{2^k} = I$ and
$$
B\sum_{i=0}^{2^k-1}A^i = B\cdot2^{k-1}(A + I) = 2^{k-1}\begin{pmatrix}1 & 2 \\ \frac{3}{2} & 3\end{pmatrix}.
$$
So the full answer is
$$
T^{2^k} = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 2^{k-1} & 2^k & 1 & 0 \\
3\cdot 2^{k-2} & 3\cdot 2^{k-1} & 0 & 1\end{pmatrix}.
$$
