TITLE: What are some mathematically interesting computations involving matrices?
QUESTION [65 upvotes]: 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.

REPLY [3 votes]: Since this got reopened again, here is a nice matrix product evaluation due to Bill Gosper.
Letting $\mathbf A_0=\mathbf I$, and evaluating the recurrence
$$\mathbf A_{k+1}=\mathbf A_k\begin{pmatrix}-\frac{k+1}{2(2k+3)}&\frac5{4(k+1)^2}\\0&1\end{pmatrix}$$
yields a sequence of matrices that converge to $\mathbf A_\infty=\begin{pmatrix}0&\zeta(3)\\0&1\end{pmatrix}$, where $\zeta(3)$ is Apéry's constant.
This can be further generalized to the evaluation of odd zeta values, by considering instead the matrix
$$\begin{pmatrix}-\frac{k+1}{2(2k+3)}&\frac1{(2k+2)(2k+3)}&&&\frac1{(k+1)^{2n}}\\&\ddots&\ddots&&\vdots\\&&\ddots&\frac1{(2k+2)(2k+3)}&\frac1{(k+1)^4}\\&&&-\frac{k+1}{2(2k+3)}&\frac5{4(k+1)^2}\\&&&&1\\\end{pmatrix}$$
which when used in the iteration will yield $\zeta(3),\zeta(5),\dots,\zeta(2n+1)$ in the last column.