Cramer's formulas for solving systems of linear equations

Consider a system of n linear equations with n unknowns x₁, x₂, ..., x_n, that is, a system in which the number of equations is equal to the number of unknowns.


The determinant of this system is called the determinant of the matrix of its coefficients


Let us replace any column, for example, the j-th, with a column b of free terms in the determinant
The determinant obtained in this way is denoted by


If the determinant of the system
then this system is compatible and has a unique solution, which is found using Cramer formulas (Cramer's rule):

Since the calculation of determinants of the 4th and higher orders is a rather cumbersome procedure, finding the roots of a system of linear equations using Cramer's formulas is appropriate for systems of two or three equations. Let us write out formulas for calculating determinants of the second and third orders.

The determinant of the second order


The determinant of the third order


Let's consider the application of Cramer's formulas using examples.

Examples of solving systems of linear equations using Cramer's formulas

Example 1.
Solve a system of two linear equations with two unknowns
Solution.
1) Let's calculate the determinant of this system:
The determinant of the system is equal to -9 ≠ 0, therefore, the system is compatible and has a unique solution.

2) Let's calculate two determinants that are obtained from the determinant of the system by replacing the first and second columns, respectively, with a column of free terms:

3) Using Cramer's formulas we calculate the values ​​of the unknowns:

Answer: x = 5, y = 1.

Example 2.
Solve a system of three linear equations with three unknowns
Solution.
1) Let's calculate the determinant of this system:
The determinant of the system is equal to 5 ≠ 0, therefore, the system is compatible and has a unique solution.

2) Let's calculate 3 determinants, which are obtained from the determinant of the system by replacing the first, second and third columns, respectively, with a column of free terms:



3) Using Cramer's formulas we calculate the values ​​of the unknowns:


Answer: x = 3, y = -1, z = 1.