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 have show a simpler visualization.