Q:

What is the limit of $A^n$ where $A=\begin{bmatrix} 1/2 & \alpha \\ 0 & 1/2 \end{bmatrix}$?

For matrix $A=\begin{bmatrix} 1/2 & \alpha \\ 0 & 1/2 \end{bmatrix}$, determine 
$$
\lim_{n \rightarrow \infty} A^n
$$

A:

$A = \dfrac{1}{2}I + \alpha N, \tag 1$
where
$N = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}; \tag 2$
note
$N^2 = 0; \tag 3$
since $NI = IN$, the binomial theorem applies to (1) and we have, via (3),
$A^k = \dfrac{1}{2^k}I + \dfrac{1}{2^{k - 1}}k\alpha N; \tag 4$
terms containing higher powers of $N$ vanish by virtue of (3).
Since
$A^k = \dfrac{1}{2^{k - 1}} \left (\dfrac{1}{2} I + k \alpha N \right ), \tag 5$
and $1/2^{k - 1}$ dominates $\alpha k$ for $k$ large, we see that
$A^k \to 0 \; \text{as} \; k \to \infty. \tag 6$
.

