Systems of Linear Equations

Consider system of m linear equations with n unknowns:

 

 

(2)

 

 

Definition. The numbers are called a solution of system (2) if substituting them into the equations, we obtain true equalities.

Definition. A system of equations (2) is said to be consistent if it has at least one solution, and it is said to be inconsistent if it has no solutions.

Definition. A system is called determined if it has a unique solution, and it is called undetermined if it has many solutions.

For example, the system of equations

has no solutions, i.e., it is inconsistent, because its left-hand sides are equal, while the right-hand sides are different.

is consistent, but undetermined, because it has infinitely many solutions. Reducing the second equation by 3, we obtain two identical equations.

 

Consider the following system of n linear equations with n unknowns

(3)

It is required to find a solution of system (3), expressed in terms of the coefficients and the free terms , where (from 1 to n).

 

Cramer’s Rule

 

To solve system (3), we multiply the first equation by А ₁₁, the second by А _21, etc., the last equation is multiplied by А_{n 1}. Then, we sum the equations and collect similar terms:

Consider the n th-order determinant composed of the coefficients of system (3):

. (4)

 

 

The coefficient of х ₁ is the sum of the products of the element of the first column and their algebraic complements. According to property 9, it equals determinant (4).

The coefficients of the unknowns are the products of the elements of the second, third, …, n th columns by the algebraic complements of the elements of the first column; consequently, they equal zero by property 10.

The right-hand side is the product of the free terms and the algebraic complements of the elements of the first column; consequently, it equals the determinant (4) in which the first column is replaced by the column of free terms:

, .

Expressions for the other unknowns are obtained in a similar way: we multiply system (3) by the algebraic complements of the n corresponding columns

, , (5)

where is the principal determinant of the system and the х_i are the auxiliary determinants obtained from the principal one by substituting the free term column for the i th columns.

Example. Solve the system of equations

Let us evaluate the principal determinant of the system:

.

To obtain zeros in the first row, we leave the third column unchanged; multiply it by –2 and add to the first column; then multiply it by 3 and add to the second column.

Let us calculate the auxiliary determinants.  х ₁ is derived from  by replacing the first column by the free terms:

х ₂ is derived from  by replacing the second column by the free terms:

х ₃ is derived from by replacing the third column by the free terms. Zeroes are obtained in the third row by adding the second column multiplied by –4 and 4 to the first and third columns, respectively:

By Cramer's rule (5) we obtain

; ;

1. In (5), the principal determinant must be different from zero. In this case, system (3) has a unique solution.

2. If =0 and one of the auxiliary determinants does not equal zero ( x 0), then the system has no solutions at school, we would say that division by zero is not allowed.

3. If =0 and all of the auxiliary determinants equal zero ( x_i =0), then the system has infinitely many solutions.