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. 

Rotation matrices are a typical example of useful matrices in computer graphics
$${\bf R_\theta} = \begin{bmatrix}\cos(\theta)&\sin(\theta)\\-\sin(\theta)&\cos(\theta)\end{bmatrix}$$
They rotate a vector around origo by the angle $\theta$.

If you want to make it more complicated you can make them in 3 dimensions. For example to rotate an angle $\theta$ around the $x$-axis:
$${\bf R_{\theta,x}} = \begin{bmatrix}1&0&0\\0&\cos(\theta)&\sin(\theta)\\0&-\sin(\theta)&\cos(\theta)\end{bmatrix}$$

Even a bit more complicated are the affine transformations, in 2D you can make one like this:
$${\bf A_{r,\theta,x_0,y_0}} = \begin{bmatrix}r\cos(\theta)&r\sin(\theta)&x_0\\-r\sin(\theta)&r\cos(\theta)&y_0\\0&0&1\end{bmatrix}, {\bf v} = \begin{bmatrix}x\\y\\1\end{bmatrix}$$
which with matrix multiplication scales the current vector ($x$ and $y$ in $\bf v$) a factor $r$, rotates it with angle $\theta$ and then adds (translates by) $[x_0,y_0]^T$.
