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. 

Here's the polynomial derivative matrix (where $n$ is the degree of the polynomial):
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}\newcommand{\i}{\mathrm{i}}\newcommand{\text}[1]{\mathrm{#1}}\newcommand{\root}[2][]{^{#2}\sqrt[#1]} \newcommand{\derivative}[3]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\abs}[1]{\left\vert\,{#1}\,\right\vert}\newcommand{\x}[0]{\times}\newcommand{\summ}[3]{\sum^{#2}_{#1}#3}\newcommand{\s}[0]{\space}\newcommand{\i}[0]{\mathrm{i}}\newcommand{\kume}[1]{\mathbb{#1}}\newcommand{\bold}[1]{\mathbf{#1}}\newcommand{\italic}[1]{\mathit{#1}}\newcommand{\kumedigerBETA}[1]{\rm #1\!#1}$
$$\large \mathbf {Derivative\s matrix\s}_{(n+1)\times (n+1)}=\begin{bmatrix}
0 & 1 & 0 & 0 & \cdots & 0 \\
0 & 0 & 2 & 0 & \cdots & 0 \\
0 & 0 & 0 & 3 & \cdots & 0 \\
0 & 0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0
\end{bmatrix}
$$
Let me do a sample calculation:

Question: What's $$\derivative{}{}{x}\left(2x^3+5x^2+7\right)$$

Find the derivative matrix and multiply it with coefficients of the polynomial.
$$\mathbf {Answer}=\begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 3\\
0 & 0 & 0 & 0\\
\end{bmatrix}
\begin{bmatrix}
7\\
0\\
5\\
2\\
\end{bmatrix}=\begin{bmatrix}
0\\
10\\
6\\
0\\
\end{bmatrix}\Rightarrow6x^2+10x
$$
And integral calculation (where $n$ is the degree of the polynomial, again)!
$$\large \mathbf {Integral\s matrix\s}_{(n+\bold{2})\times (n+\bold{2})}=\begin{bmatrix}
0 & 0 & 0 & 0 & \cdots & 0 \\
1 & 0 & 0 & 0 & \cdots & 0 \\
0 & \frac{1}{2} & 0 & 0 & \cdots & 0 \\
0 & 0 & \frac{1}{3} & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0
\end{bmatrix}
$$

Question: What's $$\int\left(2x^3+5x^2+7\right)\cdot\mathrm{d}x$$

$$\mathbf {Answer}=\begin{bmatrix}
0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0\\
0 & \frac{1}{2} & 0 & 0 & 0\\
0 & 0 & \frac{1}{3} & 0 & 0\\
0 & 0 & 0 & \frac{1}{4} & 0\\
\end{bmatrix}
\begin{bmatrix}
7\\
0\\
5\\
2\\
0\\
\end{bmatrix}=\begin{bmatrix}
0\\
7\\
0\\
\frac{5}{3}\\
\frac{1}{2}\\
\end{bmatrix}\Rightarrow \frac{1}{2}x^4+\frac{5}{3}x^3+7x+\mathit{constant}
$$
Warning: Don't forget $+constant$.
Inspired by: $3$Blue$1$Brown
