How to use Sarrus Rule in Finding Determinant of a 3x3 Matrix

Calculation of a 3x3 matrix determinant using Sarrus Rule.

Sarrus' Rule or Sarrus' Scheme is a method and a memorization scheme to compute the determinant of a matrix. It is named after the French mathematician Pierre Frédéric Sarrus. There are four Sarrus’s rules or Sarrus’ schemes to compute the determinant of a matrix.

Sarrus’ scheme 1:

Consider an arbitrary 3x3 matrix A=[aij]. The determinant of A is defined as follows:  

Observe that there are six products, each product consisting of three elements of the original matrix. Three of the products are plus-labeled (keep their sign) and three of the products are minus-labeled (change their sign).

The diagram below may help to remember the above six products in det (A).  Write first the two columns of matrix to the right of the 3rd column, so we have a total of 5 columns. That is, the determinant is equal to the sum of the products of the elements along the three plus-labeled arrows minus the negatives of the products of the elements along the three minus-labeled arrows. 

Example 1. Find the determinant of matrix A using Sarrus Scheme 1.

Solution:

Example 2. Find the determinant of matrix B using Sarrus Scheme 1.

Solution:

Sarrus’ scheme 2:

    Write out the 2 columns (2^nd and 3^rd column) of the matrix to the left of the matrix, so that you have a total of 5 columns. Then add the products of the diagonals going from top to bottom and subtract the products of the diagonals going from bottom to top. These yields

Example 1. Find the determinant of matrix A using Sarrus Scheme 2.

Solution:

Example 2. Find the determinant of matrix B using Sarrus Scheme 2.

Solution:

Sarrus’ scheme 3:

Write out the first 2 rows of the matrix to the bottom of the 3^rd row, so that you have a total of 5 rows. Then add the products of the diagonals going from top to bottom and subtract the products of the diagonals going from bottom to top. These yields

Example 1. Find the determinant of matrix A using Sarrus Scheme 3.

Solution:

Example 2. Find the determinant of matrix B using Sarrus Scheme 3.

Solution:

Sarrus’ scheme 4:

        Write out the 2 rows (2^nd and 3^rd rows) of the matrix to the above of the 1^st row, so that you have a total of 5 rows. Then add the products of the diagonals going from top to bottom and subtract the products of the diagonals going from bottom to top. These yields

Example 1. Find the determinant of matrix A using Sarrus Scheme 4.

Solution:

Example 2. Find the determinant of matrix B using Sarrus Scheme 4.

Solution:

To find more ways on how to calculate the determinant of a 3x3 matrix, click the following:

Row Echelon Form
Triangle's Rule
Co-Factor Expansion along the Column
Co-Factor Expansion along the Row