Sarrus' Rule. Determinant and Sarrus' Rule

Every square matrix can be associated with a number, which is obtained from calculations carried out between the elements of this matrix. This number is called the determinant.

The order of the square matrix determines the best method for calculating its determinant. For matrices of order 2, for example, it is enough to find the difference between the product of the elements of the main diagonal and the product of the elements of the secondary diagonal. For 3x3 matrices, we can apply the Sarrus rule or even the Laplace's Theorem. It is worth remembering that the latter can also be used to calculate determinants of square matrices of order greater than 3. In specific cases, the calculation of the determinant can be simplified by just a few determinant properties.

To understand how the determinant is calculated with the Sarrus rule, consider the following matrix A of order 3:

Representation of an order 3 matrix

Initially, the first two columns are repeated to the right of matrix A:

We must repeat the first two columns to the right of the matrix

Then the elements of the main diagonal are multiplied. This process must also be done with the diagonals that are to the right of the main diagonal so that it is possible add the products of these three diagonals:

det A_for = The₁₁.The₂₂.The₃₃ + the₁₂.The₂₃.The₃₁ + the₁₃.The₂₁.The₃₂

We must add the products of the main diagonals

The same process must be carried out with the secondary diagonal and the other diagonals to its right. However, it is necessary subtract the products found:

det A_s = - a₁₃.The₂₂.The₃₁ - a₁₁.The₂₃.The₃₃ - a₁₂.The₂₁.The₃₃

We must subtract the products from the secondary diagonals

Joining the two processes, it is possible to find the determinant of matrix A:

det A = det A_for + det A_s

det A = The₁₁.The₂₂.The₃₃ + the₁₂.The₂₃.The₃₁ + the₁₃.The₂₁.The₃₂- a₁₃.The₂₂.The₃₁ - a₁₁.The₂₃.The₃₃ - a₁₂.The₂₁.The₃₃

Representation of the application of the Sarrus Rule

Now see the calculation of the determinant of the following matrix B of order 3x3:

Calculation of the determinant of matrix B using Sarrus' Rule

Using Sarrus' rule, the calculation of the determinant of matrix B will be done as follows:

Applying Sarrus' Rule to Find the Determinant of Matrix B

det B = B₁₁.B₂₂.B₃₃ + b₁₂.B₂₃.B₃₁ + b₁₃.B₂₁.B₃₂- B₁₃.B₂₂.B₃₁ - B₁₁.B₂₃.B₃₃ - B₁₂.B₂₁.B₃₃

det B = 1.3.2 + 5.0.4 + (–2).8.(–1) – (–2).3.4 – 1.0.(–1) – 5.8.2

det B = 6 + 0 + 16 – (–24) – 0 – 80

det B = 22– 56

det B = – 34

Therefore, by Sarrus' Rule, the determinant of matrix B is – 34.

By Amanda Gonçalves
Graduated in Mathematics

Source: Brazil School - https://brasilescola.uol.com.br/matematica/regra-sarrus.htm