Pierre Frédéric Sarrus and his rule

Pierre Frédéric Sarrus was a French mathematician; he was born in 1798 and died in 1861. Teaching at the Strasbourg University, he kept himself busy with both math and astronomy (as so often those days). He is mostly known for his “Rule of Sarrus”, giving a easy solution to 3×3 matrices in linear algebra. It’s a handy way to memorize how to cope with these determinants. Nowadays computer programs or apps will come to the rescue, but there’s nothing wrong with some old-fashioned handwork.

You have a matrix

    \[\begin{bmatrix}a&d&g\\b&e&h\\c&f&i\end{bmatrix}\]

Let’s calculate the determinant:

    \[a\begin{vmatrix}e&h\\f&i\end{vmatrix}-b\begin{vmatrix}d&g\\f&i\end{vmatrix}+c\begin{vmatrix}d&g\\e&h\end{vmatrix}\]

    \[=a(ei-fh)-b(di-fg)+c(dh-eg)=aei-afh-bdi+bfg+cdh-ceg\]

Collect the positive and negative terms:

    \[={\color{red}aei}+{\color{blue}bfg}+{\color{green}cdh}-ceg-afh-bdi\]

Now look at the matrix again and add the top two rows at the bottom. You can see all pluses running diagonally from left up to right down (colored) and all minuses from left down to right up:

    \[\begin{vmatrix}{\color{red}a}&d&g\\{\color{blue}b}&{\color{red}e}&h\\{\color{green}c}&{\color{blue}f}&{\color{red}i}\\a&{\color{green}d}&{\color{blue}g}\\b&e&{\color{green}h}\end{vmatrix}\]