Cramer’s Rule

Cramer’s rules are named after Gabriel Cramer, who invented the Cramer’s rule in the 1750s to find the solution to a linear system containing the same number of variables and the same number of equations. This method is generally used to calculate the value of any particular variable without finding out the other variables or solving the entire system. It is one of the most important rules in mathematics. Cramer’s rule involves the application of matrices and determinants for solving linear equations and finding unique solutions to the system.

Cramer’s Rule by two variables 

Consider a system of two variables

a1x + b1y = c1 

a2x + b2y = c2 

Step 1. Convert the system into a matrix form AX = B, where A is the coefficient matrix, X is the variable matrix, B is the constant matrix. (The coefficient matrix is a matrix that contains only the coefficients of variables in a given set of linear equations. The variable matrix, as the name suggests, has only the variables. The constant matrix contains the constant terms given in the system of linear equations.) 

Step 2. Find the determinant (D) of system A, find the determinants  Dx and Dy, where Dx is the determinant of system A when the elements of B replace the first column, Dy is the determinant of system B when the elements of B replace the second column.

Note: Determinant of the system A must not be zero ( D ≠ 0 ).

Step 3. To find out the value of the variables x and y, we would do the following steps: 

 x = Dx/D

 y = Dy/D 

Example: 

Cramer’s Rule by three variables

Let’s consider a system of three variables x, y, and z.

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂ 

a₃x + b₃y + c₃z = d₃

Step-1: Convert the system into this in matrix form  AX = B, where A is the coefficient matrix, X is the variable matrix, B is the constant matrix.

Step-2: Find the determinant (D) of the system A, find the determinants  Dx, Dy, and Dz where Dx is the determinant of system A when the elements of B replace the first column, Dy is the determinant of system A when the second column is replaced by the elements of B and Dz is the element determinant of system A when elements of B replace the third column. 

Step-3: Find the values of the variables x, y, and z (when D ≠ 0)

x = Dₓ/D

y = Dᵧ/D

z = Dz/D 

Example :- 

4x + 2y + 2z = 6

2x – 2y – 2z = 0

2x + 4y + 2z = 0 

• First, we will convert the linear system into matrix form AX = B

• Find the determinants D, Dx, Dy, and Dz using the properties of the determinant.

D=                              

i) D = 4[ (-2)2 – (-2)(4) ] – 2[ 2(2) – (-2)2 ] + 2[ 2(4) – (-2)(2) ]

       = 4(-4 -(-8)) -2( 4+4) +2( 8+4)

       = 4(4) -2(8) +2(12) 

Dx =

       = 16 – 16 + 24

       = 24   

ii) Dx = 6[ (-2)2 – (-2)(4) ] – 2[ 0(2) – 0(-2) ] + 2[ 0(4) – (0)(-2) ]

        = 6(-4 +8) -2 (0) + 2(0)

Dy = 

        = 6(4) +0 + 0

        = 24 

iii) Dy =  4[ (0)(-2) – (-2)(0) ] – 6[ 2(2) – (-2)(2) ] + 2[ (2)(0) – (2)(0) ]

         = 4(0) -6(4 + 4) + 2(0)

 Dz = 

         = 0 – 48 +0

         = -48

iv) Dz = 4[ (-2)(0) – (4)(0) ] – 2[ (2)(0) – (2)(0) ] + 6[ (2)(4) – (-2)(2) ]

     = 4(0) -2(0) + 6(8+4)

     = 0 + 0 +72

     = 72

x = Dx/D = 24/24 = 1

y = Dy/D = -48/24 = -2

z = Dz/D = 72/24 =3

So, we can see the value of x, y, and z is 1, -2, and 3, respectively.

Cramer’s Rule Conditions

When employing Cramer’s rule by three variables to solve the given system of equations, there are a few requirements that must be met. Cramer’s rule is only applicable when a square matrix forms the linear equations. These are a few of the more notable ones: 

Because D must be in the denominator to discover the values of unknowns, Cramer’s rule fails in the case of a system of equations in which D = 0 because the values of unknowns become undefinable.

In addition, when D = 0, there will be two alternatives for which to choose:

i) It is possible that the system will not provide a solution.

ii) the system may have an endless number of solutions.

As a result, we may conclude that either at least one of the numerator determinants is a 0 (which indicates an unlimited number of solutions) or that none of the numerator determinants is a 0 (which means no solution).

If D is less than zero, we may claim that AX = B has a single solution.

Consequently, Cramer’s rule aids us in determining whether a particular system has “no solution” or “an unlimited number of solutions” by using the determinants we compute to apply the rule to the system under consideration.

Conclusion

Cramer’s rule by three variables is used to find the unique solution to linear equations containing three variables. A significant requirement is that the coefficient matrix should not have a determinant value equal to 0. Using this method, we can calculate the value of any one variable without calculating the values of all three variables. Hence, it makes our work easy. We use the concept of determinants to apply this rule.