There is this famous Rule of Sarrus for calculating the determinant of a 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 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 matrix. My first idea was to you Laplace’s formula for the first column. That is, I use the formula
in which is the determinant of the sub-matrix obtained by removing the first column and the th row of . For the entry 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 , and 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.