Cramer’s rule

Cramer’s rule is a method for solving systems of equations that is based on matrices.

To find solution of certain linear equations in n unknowns we can use Cramer’s rule.

If AX = B is a system of n linear equations in n unknowns such that determinant (A)≠ 0, then the system has a unique solution. This solution is

x₁ =  

x2 = 

………

xn = 

where Aj is the matrix obtained by replacing the entries in the jth column of A by the entries in the matrix

B =

How to do it?

Determinants can be used to solve a linear system of equations using Cramer’s Rule.

Cramer’s Rule for Two Equations in Two Variables

Given the system  ,a₁x + b₁y = c₁

a₂x + b₂y = c₂

This system has the unique solution x = , y =

where D = ,Dx = ,Dy =  ,D ≠0

When solving a system of equations using Cramer’s Rule, remember the following:

• Three different determinants are used to find x and y. The determinants in the denominators are identical.
• The elements of D, the determinant in the denominator, are the coefficients of the variables in the system; coefficients of x in the first column and coefficients of y in the second column.

D =

D_x, the determinant in the numerator of x, is obtained by replacing the x-coefficients, a₁ and a₂, in D with the constants from the right sides of the equations,c₁ and c₂ .

D = , Dx =

 D_y, the determinant in the numerator for y, is obtained by replacing the y-coefficients, b₁ and b₂, in D with the constants from the right side of the equation, c₁ and c₂.

D = , Dy = 

Example: Use Cramer’s Rule to solve the system:

5x – 4y = 2

6x – 5y = 1

Solution:

D = = = (5)(-5) - (6)(-4) = - 25 + 24 = -1

Dx = = = (2)(-5) - (1)(-4) = - 10 + 4 = -6

Dy =  = = (5)(1) - (6)(2) = 5 - 12 = -7

From Cramer’s Rule, we have x = = -6 / -1 = 6 and y = = -1 / -1 = 1

The solution is (6,7).

Note: Cramer’s Rule does not apply if D = 0. When D = 0, the system is either inconsistent or dependent.

Cramer’s Rule can be generalized to systems of linear equations with more than two variables. Suppose we are given a system with the determinant of the coefficient matrix D. Let D_x denote the determinant of the matrix obtained by replacing the column containing the coefficients of "n" with the constants from the right sides of the equations. Then we have the following result:

If a linear system of equations with variables x, y, z, . . . has a unique solution given by the formulas

x = , y = , z =