There is this famous Rule of Sarrus for calculating the determinant of a ${3\times 3}$ matrix. The nice thing is, that one can remember it quite quickly with the help of a simple picture which I cite from the Wikipedia page:

Being at a university which has a lot of engineers who need to learn mathematics, there are plenty of students who learn about determinants and also learn the Rule of Sarrus. A quite amusing thing which happens frequently in exams is the following (at least colleagues told me so): If you ask the students to calculate the determinant of a ${4\times 4}$ matrix there will be a number of people adopting the Rule of Sarrus without thinking, ending up with the sum of eight products. You will also find interesting discussions if you google “Sarrus 4×4”. Of course, the Rule of Sarrus can not work in this simple manner since, according to the Leibniz formula for determinants, you need a sum of 24 products.

Well, I thought it could be nice to have an illustration of the 24 “paths” along which you should take products in a ${4\times 4}$ matrix. My first idea was to you Laplace’s formula for the first column. That is, I use the formula

$\displaystyle \det(A) = \sum_{i=1}^4 (-1)^{i+1}a_{i,1}M_{i,1}$

in which ${M_{i,1}}$ is the determinant of the ${3\times 3}$ sub-matrix obtained by removing the first column and the ${i}$th row of ${A}$. For the entry ${a_{1,1}}$ this gives this picture:

Here, the green lines indicate products which get the sign ${+}$ and red lines indicate products which get the sign ${-}$. The blue dots are just for better orientation.

Similarly, the elements ${a_{2,1}}$, ${a_{3,1}}$ and ${a_{4,1}}$ lead to similar pictures:

Putting all graphics together we obtain the nice and intuitive picture

for the 4×4 Rule of Sarrus. Ok – have fun memorizing this…

Since this picture looks so ugly, one may be tempted to call the corresponding rule the “Rule of Sauron”…

P.S.: Probably there is some graph theorist somewhere who could produce a nicer picture, e.g. one which minimizes the number of crossing. Moreover, there are probably other thoughts about the Ruls of Sarrus and its interpretations for larger matrices – I would be glad to learn about them.

In a follow-up post, I show a simpler visualization.