SARRUS RULE: WHAT IT CONSISTS OF AND TYPES OF DETERMINANTS - CULTURE VOCABULARY - 2025

The rule Sarrus is used to calculate the result of 3 × 3 determinants. These are used to solve linear equations and find out if they are compatible.

Compatible systems make it easier to obtain the solution. They are also used to determine whether sets of vectors are linearly independent and to form the basis of the vector space.

These applications are based on the invertibility of the matrices. If a matrix is ​​regular, its determinant is different from 0. If it is singular, its determinant is equal to 0. Determinants can only be calculated in square matrices.

To compute matrices of any order, Laplace's theorem can be used. This theorem allows us to simplify matrices of high dimensions, in sums of small determinants that we decompose from the main matrix.

It states that the determinant of a matrix is ​​equal to the sum of the products of each row or column, times the determinant of its adjoint matrix.

This reduces the determinants so that a determinant of degree n becomes n determinants of n-1. If we apply this rule successively, we can obtain determinants of dimension 2 (2 × 2) or 3 (3 × 3), where its calculation is much easier.

Sarrus rule

Pierre Frederic Sarrus was a 19th century French mathematician. Most of his mathematical treatises are based on methods of solving equations and the calculus of variations, within numerical equations.

In one of his treatises, he solved one of the most complex riddles in mechanics. To solve the problems of the articulated pieces, Sarrus introduced the transformation of alternative rectilinear movements, in uniform circular movements. This new system is known as the Sarrus mechanism.

The research that gave this mathematician most fame was in which he introduced a new method of calculating determinants, in the article "Nouvelles méthodes pour la résolution des équations" (New method for solving equations), which was published in the year 1833. This way of solving linear equations is known as Sarrus's rule.

The Sarrus rule allows calculating the determinant of a 3 × 3 matrix, without the need to use Laplace's theorem, introducing a much simpler and more intuitive method. In order to check the value of Sarrus's rule, we take any matrix of dimension 3:

The calculation of its determinant would be carried out using the product of its main diagonals, subtracting the product of the inverse diagonals. This would be as follows:

Sarrus' rule allows us to obtain a much easier vision when calculating the diagonals of the determinant. It would be simplified by adding the first two columns to the back of the matrix. In this way, it is more clearly seen which are its main diagonals and which are the inverse ones, for the calculation of the product.

Through this image we can see the application of Sarrus's rule, we include row 1 and 2, below the graphic representation of the initial matrix. In this way, the main diagonals are the three diagonals that appear first.

The three reverse diagonals, in turn, are those that appear first at the back.

In this way, the diagonals appear in a more visual way, without complicating the resolution of the determinant, trying to find out which elements of the matrix belong to each diagonal.

As it appears in the image, we choose the diagonals and calculate the resulting product of each function. The diagonals that appear in blue are those that add up. To the sum of these, we subtract the value of the diagonals that appear in red.

To make compression easier, we can use a numerical example, instead of using algebraic terms and subterms.

If we take any 3 × 3 matrix, for example:

To apply Sarrus's rule, and solve it in a more visual way, we should include row 1 and 2, as row 4 and 5 respectively. It is important to keep row 1 in the 4th position, and row 2 in the 5th position. Since if we exchange them, the Sarrus Rule will not be effective.

To calculate the determinant, our matrix would be as follows:

To continue with the calculation, we will multiply the elements of the main diagonals. Descendants starting from the left will have a positive sign; while the inverse diagonals, which start from the right, have a negative sign.

In this example, the blue ones would have a positive sign and the red ones with a negative sign. The final calculation of the Sarrus Rule would look like this:

Types of determinants

Determinant of dimension 1

If the dimension of the matrix is ​​1, the matrix looks like this: A = (a)

Therefore, its determinant would be as follows: det (A) = -A- = a

In summary, the determinant of matrix A is equal to the absolute value of matrix A, which in this case is a.

Determinant of dimension 2

If we pass to matrices of dimension 2, we obtain matrices of the type:

Where its determinant is defined as:

The resolution of this determinant is based on the multiplication of its main diagonal, subtracting the product of its inverse diagonal.

As a mnemonic, we can use the following diagram to remember its determinant:

Determinant of dimension 3

If the dimension of the matrix is ​​3, the resulting matrix would be of this type:

The determinant of this matrix would be solved through Sarrus's rule in this way:

References

1. Jenny Olive (1998) Maths: A Student's Survival Guide. Cambridge University Press.
2. Richard J. Brown (2012) 30-Second Maths: The 50 Most Mind-Expanding Theories in Mathematics. Ivy Press Limited.
3. Dave Kirkby (2004) Maths Connect. Heinemann.
4. Awol Assen (2013) A Study on the Computation of the Determinants of a 3 × 3 Matrix. Lap Lambert Academic Publishing.
5. Anthony Nicolaides (1994) Determinants & Matrices. Pass Publication.
6. Jesse Russell (2012) Rule of Sarrus.
7. M. Casteleiro Villalba (2004) Introduction to linear algebra. ESIC Editorial.