Why are these examples striking? The question is from an exercise in Gilbert Strang's Linear Algebra and its Applications.

The powers $A^k$ approach zero if all $|\lambda_i|<1$, and they blow up if any $|\lambda_i|>1$. Peter Lax gives four striking examples in his book Linear Algebra.
  $$A = \left(
  \begin{array}{cc}
     3& 2 \\
     1& 4 \\
  \end{array}
\right)\qquad B = 
\left(
  \begin{array}{cc}
     3 & 2  \\
     -5 & -3  \\
  \end{array}
\right)\qquad C = 
\left(
  \begin{array}{cc}
     5& 7 \\
     -3& -4  \\
  \end{array}
\right)\qquad D = 
\left(
  \begin{array}{cc}
     5& 6.9  \\
     -3& -4 \\
  \end{array}
\right)$$
  $$\|A^{1024}\|>10^{700}\qquad B^{1024}=I\qquad C^{1024}=-C\qquad \|D^{1024}\|<10^{-78}$$
  Find the eigenvalues $\lambda=e^{i\theta}$ of $B$ and $C$ to show that $B^4=I$ and $C^3=-I$.

Here is my question:
Why are these examples so special? Is it because that all of them contain the number "1024"? Or such examples are hard to construct?
 A: The examples display three different behaviors:


*
1. $\|A^k\|\to\infty$
2. $B^k$ and $C^k$ cycle with periods of $4$ and $6$ respectively, always with norm $1$
3.$\|D^k\|\to0$ 


$1024$ is just a large number that displays these behaviors strikingly.  Since $1024=0\pmod{4}$ and $1024=4\pmod{6}$, so $B^{1024}=I$ and $C^{1024}=-C$.
Peter Lax said the examples were "striking", not "special".
A: There is nothing particularly special about $1024$; it just happens to be convenient to compute matrices to $2^n$ powers because you can repeatedly square. 
What's supposed to be striking (of course this is subjective) is that $A, B, C, D$ all have small, superficially similar-looking entries, but after sufficient iteration have very different qualitative behavior; moreover, if you didn't know a lot of linear algebra, you would be hard-pressed to guess which matrices would admit which behavior just by looking at them. 
Examples $A, D$ are not hard to construct in the sense that a random matrix you choose will do one of those two things. Examples $B, C$ are more fine-tuned, but not hard to construct by hand once you understand how to construct matrices with a given characteristic polynomial. 
