System of Linear Equations in three variables using Cramer’s Rule

Cramer’s rule: In linear algebra, Cramer’s rule is an explicit formula for the solution of a system of linear equations with as many equations as unknown variables. It expresses the solution in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the column vector of the right-hand-sides of the equations. Cramer’s rule is computationally inefficient for systems of more than two or three equations.
Suppose we have to solve these equations:

a₁x + b₁y + c₁z = d₁ 
a₂x + b₂y + c₂z = d₂ 
a₃x + b₃y + c₃z = d₃

Following the Cramer’s Rule, first find the determinant values of all four matrices.

$D = \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3\\ \end{vmatrix}$

$D_1 = \begin{vmatrix} d_1 & b_1 & c_1\\ d_2 & b_2 & c_2\\ d_3 & b_3 & c_3\\ \end{vmatrix}$

$D_2 = \begin{vmatrix} a_1 & d_1 & c_1\\ a_2 & d_2 & c_2\\ a_3 & d_3 & c_3\\ \end{vmatrix}$

$D_3 = \begin{vmatrix} a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & b_3 & d_3\\ \end{vmatrix}$

There are 2 cases:
Case I : When D ≠ 0 In this case we have,
x = D1/D
y = D2/D
z = D3/D
Hence unique value of x, y, z will be obtained.
Case II : When D = 0
When at least one of D1, D2 and D3 is non zero: Then no solution is possible and hence system of equations will be inconsistent.
When D = 0 and D1 = D2 = D3 = 0: Then the system of equations will be consistent and it will have infinitely many solutions.

Example:

Consider the following system of linear equations.
[2x – y + 3z = 9], [x + y + z = 6], [x – y + z = 2]
$D = \begin{vmatrix} 2 & -1 & 3\\ 1 & 1 & 1\\ 1 & -1 & 1\\ \end{vmatrix}$ [Tex]D_1 = \begin{vmatrix} 9 & -1 & 3\\ 6 & 1 & 1\\ 2 & -1 & 1\\ \end{vmatrix} [/Tex] $D_2 = \begin{vmatrix} 2 & 9 & 3\\ 1 & 6 & 1\\ 1 & 2 & 1\\ \end{vmatrix}$ [Tex]D_3 = \begin{vmatrix} 2 & -1 & 9\\ 1 & 1 & 6\\ 1 & -1 & 2\\ \end{vmatrix} [/Tex]
[x = D₁/D = 1], [y = D₂/D = 2], [z = D₃/D = 3]

Below is the implementation.

C++

// CPP program to calculate solutions of linear
// equations using cramer's rule
#include <bits/stdc++.h>
using namespace std;
 
// This functions finds the determinant of Matrix
double determinantOfMatrix(double mat[3][3])
{
    double ans;
    ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])
          - mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])
          + mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);
    return ans;
}
 
// This function finds the solution of system of
// linear equations using cramer's rule
void findSolution(double coeff[3][4])
{
    // Matrix d using coeff as given in cramer's rule
    double d[3][3] = {
        { coeff[0][0], coeff[0][1], coeff[0][2] },
        { coeff[1][0], coeff[1][1], coeff[1][2] },
        { coeff[2][0], coeff[2][1], coeff[2][2] },
    };
    // Matrix d1 using coeff as given in cramer's rule
    double d1[3][3] = {
        { coeff[0][3], coeff[0][1], coeff[0][2] },
        { coeff[1][3], coeff[1][1], coeff[1][2] },
        { coeff[2][3], coeff[2][1], coeff[2][2] },
    };
    // Matrix d2 using coeff as given in cramer's rule
    double d2[3][3] = {
        { coeff[0][0], coeff[0][3], coeff[0][2] },
        { coeff[1][0], coeff[1][3], coeff[1][2] },
        { coeff[2][0], coeff[2][3], coeff[2][2] },
    };
    // Matrix d3 using coeff as given in cramer's rule
    double d3[3][3] = {
        { coeff[0][0], coeff[0][1], coeff[0][3] },
        { coeff[1][0], coeff[1][1], coeff[1][3] },
        { coeff[2][0], coeff[2][1], coeff[2][3] },
    };
 
    // Calculating Determinant of Matrices d, d1, d2, d3
    double D = determinantOfMatrix(d);
    double D1 = determinantOfMatrix(d1);
    double D2 = determinantOfMatrix(d2);
    double D3 = determinantOfMatrix(d3);
    printf("D is : %lf \n", D);
    printf("D1 is : %lf \n", D1);
    printf("D2 is : %lf \n", D2);
    printf("D3 is : %lf \n", D3);
 
    // Case 1
    if (D != 0) {
        // Coeff have a unique solution. Apply Cramer's Rule
        double x = D1 / D;
        double y = D2 / D;
        double z = D3 / D; // calculating z using cramer's rule
        printf("Value of x is : %lf\n", x);
        printf("Value of y is : %lf\n", y);
        printf("Value of z is : %lf\n", z);
    }
    // Case 2
    else {
        if (D1 == 0 && D2 == 0 && D3 == 0)
            printf("Infinite solutions\n");
        else if (D1 != 0 || D2 != 0 || D3 != 0)
            printf("No solutions\n");
    }
}
 
// Driver Code
int main()
{
 
    // storing coefficients of linear equations in coeff matrix
    double coeff[3][4] = {
        { 2, -1, 3, 9 },
        { 1, 1, 1, 6 },
        { 1, -1, 1, 2 },
    };
 
    findSolution(coeff);
    return 0;
}

Java

// Java program to calculate solutions of linear
// equations using cramer's rule
class GFG
{
 
// This functions finds the determinant of Matrix
static double determinantOfMatrix(double mat[][])
{
    double ans;
    ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])
        - mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])
        + mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);
    return ans;
}
 
// This function finds the solution of system of
// linear equations using cramer's rule
static void findSolution(double coeff[][])
{
    // Matrix d using coeff as given in cramer's rule
    double d[][] = {
        { coeff[0][0], coeff[0][1], coeff[0][2] },
        { coeff[1][0], coeff[1][1], coeff[1][2] },
        { coeff[2][0], coeff[2][1], coeff[2][2] },
    };
     
    // Matrix d1 using coeff as given in cramer's rule
    double d1[][] = {
        { coeff[0][3], coeff[0][1], coeff[0][2] },
        { coeff[1][3], coeff[1][1], coeff[1][2] },
        { coeff[2][3], coeff[2][1], coeff[2][2] },
    };
     
    // Matrix d2 using coeff as given in cramer's rule
    double d2[][] = {
        { coeff[0][0], coeff[0][3], coeff[0][2] },
        { coeff[1][0], coeff[1][3], coeff[1][2] },
        { coeff[2][0], coeff[2][3], coeff[2][2] },
    };
     
    // Matrix d3 using coeff as given in cramer's rule
    double d3[][] = {
        { coeff[0][0], coeff[0][1], coeff[0][3] },
        { coeff[1][0], coeff[1][1], coeff[1][3] },
        { coeff[2][0], coeff[2][1], coeff[2][3] },
    };
 
    // Calculating Determinant of Matrices d, d1, d2, d3
    double D = determinantOfMatrix(d);
    double D1 = determinantOfMatrix(d1);
    double D2 = determinantOfMatrix(d2);
    double D3 = determinantOfMatrix(d3);
    System.out.printf("D is : %.6f \n", D);
    System.out.printf("D1 is : %.6f \n", D1);
    System.out.printf("D2 is : %.6f \n", D2);
    System.out.printf("D3 is : %.6f \n", D3);
 
    // Case 1
    if (D != 0)
    {
        // Coeff have a unique solution. Apply Cramer's Rule
        double x = D1 / D;
        double y = D2 / D;
        double z = D3 / D; // calculating z using cramer's rule
        System.out.printf("Value of x is : %.6f\n", x);
        System.out.printf("Value of y is : %.6f\n", y);
        System.out.printf("Value of z is : %.6f\n", z);
    }
     
    // Case 2
    else
    {
        if (D1 == 0 && D2 == 0 && D3 == 0)
            System.out.printf("Infinite solutions\n");
        else if (D1 != 0 || D2 != 0 || D3 != 0)
            System.out.printf("No solutions\n");
    }
}
 
// Driver Code
public static void main(String[] args)
{
    // storing coefficients of linear
    // equations in coeff matrix
    double coeff[][] = {{ 2, -1, 3, 9 },
                        { 1, 1, 1, 6 },
                        { 1, -1, 1, 2 }};
    findSolution(coeff);
    }
}
 
// This code is contributed by PrinciRaj1992

Python3

# Python3 program to calculate
# solutions of linear equations
# using cramer's rule
 
# This functions finds the
# determinant of Matrix
def determinantOfMatrix(mat):
 
    ans = (mat[0][0] * (mat[1][1] * mat[2][2] -
                        mat[2][1] * mat[1][2]) -
           mat[0][1] * (mat[1][0] * mat[2][2] -
                        mat[1][2] * mat[2][0]) +
           mat[0][2] * (mat[1][0] * mat[2][1] -
                        mat[1][1] * mat[2][0]))
    return ans
 
# This function finds the solution of system of
# linear equations using cramer's rule
def findSolution(coeff):
 
    # Matrix d using coeff as given in
    # cramer's rule
    d = [[coeff[0][0], coeff[0][1], coeff[0][2]],
         [coeff[1][0], coeff[1][1], coeff[1][2]],
         [coeff[2][0], coeff[2][1], coeff[2][2]]]
     
    # Matrix d1 using coeff as given in
    # cramer's rule
    d1 = [[coeff[0][3], coeff[0][1], coeff[0][2]],
          [coeff[1][3], coeff[1][1], coeff[1][2]],
          [coeff[2][3], coeff[2][1], coeff[2][2]]]
     
    # Matrix d2 using coeff as given in
    # cramer's rule
    d2 = [[coeff[0][0], coeff[0][3], coeff[0][2]],
          [coeff[1][0], coeff[1][3], coeff[1][2]],
          [coeff[2][0], coeff[2][3], coeff[2][2]]]
     
    # Matrix d3 using coeff as given in
    # cramer's rule
    d3 = [[coeff[0][0], coeff[0][1], coeff[0][3]],
          [coeff[1][0], coeff[1][1], coeff[1][3]],
          [coeff[2][0], coeff[2][1], coeff[2][3]]]
 
    # Calculating Determinant of Matrices
    # d, d1, d2, d3
    D = determinantOfMatrix(d)
    D1 = determinantOfMatrix(d1)
    D2 = determinantOfMatrix(d2)
    D3 = determinantOfMatrix(d3)
     
    print("D is : ", D)
    print("D1 is : ", D1)
    print("D2 is : ", D2)
    print("D3 is : ", D3)
 
    # Case 1
    if (D != 0):
       
        # Coeff have a unique solution.
        # Apply Cramer's Rule
        x = D1 / D
        y = D2 / D
         
        # calculating z using cramer's rule
        z = D3 / D 
         
        print("Value of x is : ", x)
        print("Value of y is : ", y)
        print("Value of z is : ", z)
 
    # Case 2
    else:
        if (D1 == 0 and D2 == 0 and
            D3 == 0):
            print("Infinite solutions")
        elif (D1 != 0 or D2 != 0 or
              D3 != 0):
            print("No solutions")
 
# Driver Code
if __name__ == "__main__":
 
    # storing coefficients of linear
    # equations in coeff matrix
    coeff = [[2, -1, 3, 9],
             [1, 1, 1, 6],
             [1, -1, 1, 2]]
 
    findSolution(coeff)
 
# This code is contributed by Chitranayal

C#

// C# program to calculate solutions of linear
// equations using cramer's rule
using System;
 
class GFG
{
 
// This functions finds the determinant of Matrix
static double determinantOfMatrix(double [,]mat)
{
    double ans;
    ans = mat[0,0] * (mat[1,1] * mat[2,2] - mat[2,1] * mat[1,2])
        - mat[0,1] * (mat[1,0] * mat[2,2] - mat[1,2] * mat[2,0])
        + mat[0,2] * (mat[1,0] * mat[2,1] - mat[1,1] * mat[2,0]);
    return ans;
}
 
// This function finds the solution of system of
// linear equations using cramer's rule
static void findSolution(double [,]coeff)
{
    // Matrix d using coeff as given in cramer's rule
    double [,]d = {
        { coeff[0,0], coeff[0,1], coeff[0,2] },
        { coeff[1,0], coeff[1,1], coeff[1,2] },
        { coeff[2,0], coeff[2,1], coeff[2,2] },
    };
     
    // Matrix d1 using coeff as given in cramer's rule
    double [,]d1 = {
        { coeff[0,3], coeff[0,1], coeff[0,2] },
        { coeff[1,3], coeff[1,1], coeff[1,2] },
        { coeff[2,3], coeff[2,1], coeff[2,2] },
    };
     
    // Matrix d2 using coeff as given in cramer's rule
    double [,]d2 = {
        { coeff[0,0], coeff[0,3], coeff[0,2] },
        { coeff[1,0], coeff[1,3], coeff[1,2] },
        { coeff[2,0], coeff[2,3], coeff[2,2] },
    };
     
    // Matrix d3 using coeff as given in cramer's rule
    double [,]d3 = {
        { coeff[0,0], coeff[0,1], coeff[0,3] },
        { coeff[1,0], coeff[1,1], coeff[1,3] },
        { coeff[2,0], coeff[2,1], coeff[2,3] },
    };
 
    // Calculating Determinant of Matrices d, d1, d2, d3
    double D = determinantOfMatrix(d);
    double D1 = determinantOfMatrix(d1);
    double D2 = determinantOfMatrix(d2);
    double D3 = determinantOfMatrix(d3);
    Console.Write("D is : {0:F6} \n", D);
    Console.Write("D1 is : {0:F6} \n", D1);
    Console.Write("D2 is : {0:F6} \n", D2);
    Console.Write("D3 is : {0:F6} \n", D3);
 
    // Case 1
    if (D != 0)
    {
        // Coeff have a unique solution. Apply Cramer's Rule
        double x = D1 / D;
        double y = D2 / D;
        double z = D3 / D; // calculating z using cramer's rule
        Console.Write("Value of x is : {0:F6}\n", x);
        Console.Write("Value of y is : {0:F6}\n", y);
        Console.Write("Value of z is : {0:F6}\n", z);
    }
     
    // Case 2
    else
    {
        if (D1 == 0 && D2 == 0 && D3 == 0)
            Console.Write("Infinite solutions\n");
        else if (D1 != 0 || D2 != 0 || D3 != 0)
            Console.Write("No solutions\n");
    }
}
 
// Driver Code
public static void Main()
{
    // storing coefficients of linear
    // equations in coeff matrix
    double [,]coeff = {{ 2, -1, 3, 9 },
                        { 1, 1, 1, 6 },
                        { 1, -1, 1, 2 }};
    findSolution(coeff);
    }
}
 
// This code is contributed by 29AjayKumar

Javascript

<script>
// Javascript program to calculate solutions of linear
// equations using cramer's rule
 
// This functions finds the determinant of Matrix
function determinantOfMatrix(mat)
{
    let ans;
    ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])
        - mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])
        + mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);
    return ans;
     
}
 
// This function finds the solution of system of
// linear equations using cramer's rule
function findSolution(coeff)
{
// Matrix d using coeff as given in cramer's rule
    let d = [[coeff[0][0], coeff[0][1], coeff[0][2]],
         [coeff[1][0], coeff[1][1], coeff[1][2]],
         [coeff[2][0], coeff[2][1], coeff[2][2]]];
     
    // Matrix d1 using coeff as given in cramer's rule
    let d1 = [[coeff[0][3], coeff[0][1], coeff[0][2]],
          [coeff[1][3], coeff[1][1], coeff[1][2]],
          [coeff[2][3], coeff[2][1], coeff[2][2]]];
           
    // Matrix d2 using coeff as given in cramer's rule     
    let d2 = [[coeff[0][0], coeff[0][3], coeff[0][2]],
          [coeff[1][0], coeff[1][3], coeff[1][2]],
          [coeff[2][0], coeff[2][3], coeff[2][2]]];
     
    // Matrix d3 using coeff as given in cramer's rule
    let d3 = [[coeff[0][0], coeff[0][1], coeff[0][3]],
          [coeff[1][0], coeff[1][1], coeff[1][3]],
          [coeff[2][0], coeff[2][1], coeff[2][3]]];
     
    // Calculating Determinant of Matrices d, d1, d2, d3
    let D = determinantOfMatrix(d);
    let D1 = determinantOfMatrix(d1);
    let D2 = determinantOfMatrix(d2);
    let D3 = determinantOfMatrix(d3);
     
    document.write("D is :  ", D.toFixed(6)+"<br>");
    document.write("D1 is : ", D1.toFixed(6)+"<br>");
    document.write("D2 is : ", D2.toFixed(6)+"<br>");
    document.write("D3 is : ", D3.toFixed(6)+"<br>");
     
    // Case 1
    if (D != 0)
    {
        // Coeff have a unique solution. Apply Cramer's Rule
        let x = D1 / D;
        let y = D2 / D;
        let z = D3 / D; // calculating z using cramer's rule
        document.write("Value of x is : ", x.toFixed(6)+"<br>");
        document.write("Value of y is : ", y.toFixed(6)+"<br>");
        document.write("Value of z is : ", z.toFixed(6)+"<br>");
    }
      
    // Case 2
    else
    {
        if (D1 == 0 && D2 == 0 && D3 == 0)
            document.write("Infinite solutions\n");
        else if (D1 != 0 || D2 != 0 || D3 != 0)
            document.write("No solutions\n");
    }
}
 
// Driver Code
let coeff = [[2, -1, 3, 9],
             [1, 1, 1, 6],
             [1, -1, 1, 2]]
  
findSolution(coeff);
     
     
    // This code is contributed by avanitrachhadiya2155
</script>

Output

D is : -2.000000 
D1 is : -2.000000 
D2 is : -4.000000 
D3 is : -6.000000 
Value of x is : 1.000000
Value of y is : 2.000000
Value of z is : 3.000000

Time complexity: O(1)
Auxiliary space: O(1)

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Last Updated : 25 Oct, 2022

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