What are some mathematically interesting computations involving matrices? I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving matrices. 
Preferably these examples would be easy to implement in a computer program. 
For instance, suppose
$$\begin{eqnarray}
F_0&=&0\\
F_1&=&1\\
F_{n+1}&=&F_n+F_{n-1},
\end{eqnarray}$$
so that $F_n$ is the $n^{th}$ term in the Fibonacci sequence. If we set
$$A=\begin{pmatrix}
1 & 1 \\ 1 & 0 
\end{pmatrix}$$
we see that
$$A^1=\begin{pmatrix}
1 & 1 \\ 1 & 0
\end{pmatrix} =
\begin{pmatrix}
F_2 & F_1 \\ F_1 & F_0
\end{pmatrix},$$
and it can be shown that
$$
A^n =
\begin{pmatrix}
F_{n+1} & F_{n} \\ F_{n} & F_{n-1}
\end{pmatrix}.$$
This example is "interesting" in that it provides a novel way to compute the Fibonacci sequence. It is also relatively easy to implement a simple program to verify the above.
Other examples like this will be much appreciated. 

If $(a,b,c)$ is a Pythagorean triple (i.e. positive integers such that $a^2+b^2=c^2$), then
$$\underset{:=A}{\underbrace{\begin{pmatrix}
1 & -2 & 2\\ 
2 & -1 & 2\\ 
2 & -2 & 3
\end{pmatrix}}}\begin{pmatrix}
a\\ 
b\\ 
c
\end{pmatrix}$$
is also a Pythagorean triple. In addition, if the initial triple is primitive (i.e. $a$, $b$ and $c$ share no common divisor), then so is the result of the multiplication.
The same is true if we replace $A$ by one of the following matrices:
$$B:=\begin{pmatrix}
1 & 2 & 2\\ 
2 & 1 & 2\\ 
2 & 2 & 3
\end{pmatrix}
\quad \text{or}\quad 
C:=\begin{pmatrix}
-1 & 2 & 2\\ 
-2 & 1 & 2\\ 
-2 & 2 & 3
\end{pmatrix}.
$$
Taking $x=(3,4,5)$ as initial triple, we can use the matrices $A$, $B$ and $C$ to construct a tree with all primitive Pythagorean triples (without repetition) as follows:
$$x\left\{\begin{matrix}
Ax\left\{\begin{matrix}
AAx\cdots\\ 
BAx\cdots\\ 
CAx\cdots
\end{matrix}\right.\\ \\
Bx\left\{\begin{matrix}
ABx\cdots\\ 
BBx\cdots\\ 
CBx\cdots
\end{matrix}\right.\\ \\
Cx\left\{\begin{matrix}
ACx\cdots\\ 
BCx\cdots\\ 
CCx\cdots
\end{matrix}\right.
\end{matrix}\right.$$
Source: Wikipedia's page Tree of primitive Pythagorean triples.
