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Program for Gauss-Jordan Elimination Method
  • Difficulty Level : Medium
  • Last Updated : 21 Apr, 2020
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Prerequisite : Gaussian Elimination to Solve Linear Equations

Introduction : The Gauss-Jordan method, also known as Gauss-Jordan elimination method is used to solve a system of linear equations and is a modified version of Gauss Elimination Method.

It is similar and simpler than Gauss Elimination Method as we have to perform 2 different process in Gauss Elimination Method i.e.
1) Formation of upper triangular matrix, and
2) Back substitution

But in case of Gauss-Jordan Elimination Method, we only have to form a reduced row echelon form (diagonal matrix). Below given is the flow-chart of Gauss-Jordan Elimination Method.

Flow Chart of Gauss-Jordan Elimination Method :



Examples :

Input :  2y + z = 4
         x + y + 2z = 6
         2x + y + z = 7

Output :
Final Augumented Matrix is : 
1 0 0 2.2 
0 2 0 2.8 
0 0 -2.5 -3 

Result is : 2.2 1.4 1.2 

Explanation : Below given is the explanation of the above example.

  • Input Augmented Matrix is :

  • Interchanging R1 and R2, we get

  • Performing the row operation R3 <- R3 – (2*R1)

  • Performing the row operations R1 <- R1 – ((1/2)* R2) and R3 <- R3 + ((1/2)*R2)

  • Performing R1 <- R1 + ((3/5)*R3) and R2 <- R2 + ((2/5)*R3)

  • Unique Solutions are :

    C++




    // C++ Implementation for Gauss-Jordan
    // Elimination Method
    #include <bits/stdc++.h>
    using namespace std;
      
    #define M 10
      
    // Function to print the matrix
    void PrintMatrix(float a[][M], int n)
    {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j <= n; j++) 
              cout << a[i][j] << " ";
            cout << endl;
        }
    }
      
    // function to reduce matrix to reduced
    // row echelon form.
    int PerformOperation(float a[][M], int n)
    {
        int i, j, k = 0, c, flag = 0, m = 0;
        float pro = 0;
          
        // Performing elementary operations
        for (i = 0; i < n; i++)
        {
            if (a[i][i] == 0) 
            {
                c = 1;
                while ((i + c) < n && a[i + c][i] == 0) 
                    c++;            
                if ((i + c) == n) {
                    flag = 1;
                    break;
                }
                for (j = i, k = 0; k <= n; k++) 
                    swap(a[j][k], a[j+c][k]);
            }
      
            for (j = 0; j < n; j++) {
                  
                // Excluding all i == j
                if (i != j) {
                      
                    // Converting Matrix to reduced row
                    // echelon form(diagonal matrix)
                    float pro = a[j][i] / a[i][i];
      
                    for (k = 0; k <= n; k++)                 
                        a[j][k] = a[j][k] - (a[i][k]) * pro;                
                }
            }
        }
        return flag;
    }
      
    // Function to print the desired result 
    // if unique solutions exists, otherwise 
    // prints no solution or infinite solutions 
    // depending upon the input given.
    void PrintResult(float a[][M], int n, int flag)
    {
        cout << "Result is : ";
      
        if (flag == 2)     
          cout << "Infinite Solutions Exists" << endl;    
        else if (flag == 3)     
          cout << "No Solution Exists" << endl;
          
          
        // Printing the solution by dividing constants by
        // their respective diagonal elements
        else {
            for (int i = 0; i < n; i++)         
                cout << a[i][n] / a[i][i] << " ";        
        }
    }
      
    // To check whether infinite solutions 
    // exists or no solution exists
    int CheckConsistency(float a[][M], int n, int flag)
    {
        int i, j;
        float sum;
          
        // flag == 2 for infinite solution
        // flag == 3 for No solution
        flag = 3;
        for (i = 0; i < n; i++) 
        {
            sum = 0;
            for (j = 0; j < n; j++)        
                sum = sum + a[i][j];
            if (sum == a[i][j]) 
                flag = 2;        
        }
        return flag;
    }
      
    // Driver code
    int main()
    {
        float a[M][M] = {{ 0, 2, 1, 4 }, 
                         { 1, 1, 2, 6 }, 
                         { 2, 1, 1, 7 }};
                           
        // Order of Matrix(n)
        int n = 3, flag = 0;
          
        // Performing Matrix transformation
        flag = PerformOperation(a, n);
          
        if (flag == 1)     
            flag = CheckConsistency(a, n, flag);    
      
        // Printing Final Matrix
        cout << "Final Augumented Matrix is : " << endl;
        PrintMatrix(a, n);
        cout << endl;
          
        // Printing Solutions(if exist)
        PrintResult(a, n, flag);
      
        return 0;
    }

    Java




    // Java Implementation for Gauss-Jordan
    // Elimination Method
    class GFG {
          
    static int M = 10;
      
    // Function to print the matrix
    static void PrintMatrix(float a[][], int n)
    {
        for (int i = 0; i < n; i++) 
        {
            for (int j = 0; j <= n; j++) 
                System.out.print(a[i][j] + " ");
            System.out.println();
        }
    }
      
    // function to reduce matrix to reduced
    // row echelon form.
    static int PerformOperation(float a[][], int n)
    {
        int i, j, k = 0, c, flag = 0, m = 0;
        float pro = 0;
          
        // Performing elementary operations
        for (i = 0; i < n; i++)
        {
            if (a[i][i] == 0
            {
                c = 1;
                while ((i + c) < n && a[i + c][i] == 0
                    c++;         
                if ((i + c) == n) 
                {
                    flag = 1;
                    break;
                }
                for (j = i, k = 0; k <= n; k++) 
                {
                    float temp =a[j][k];
                    a[j][k] = a[j+c][k];
                    a[j+c][k] = temp;
                }
            }
      
            for (j = 0; j < n; j++) 
            {
                  
                // Excluding all i == j
                if (i != j) 
                {
                      
                    // Converting Matrix to reduced row
                    // echelon form(diagonal matrix)
                    float p = a[j][i] / a[i][i];
      
                    for (k = 0; k <= n; k++)                 
                        a[j][k] = a[j][k] - (a[i][k]) * p;             
                }
            }
        }
        return flag;
    }
      
    // Function to print the desired result 
    // if unique solutions exists, otherwise 
    // prints no solution or infinite solutions 
    // depending upon the input given.
    static void PrintResult(float a[][], int n, int flag)
    {
        System.out.print("Result is : ");
      
        if (flag == 2)     
        System.out.println("Infinite Solutions Exists"); 
        else if (flag == 3)     
        System.out.println("No Solution Exists");
          
          
        // Printing the solution by dividing constants by
        // their respective diagonal elements
        else {
            for (int i = 0; i < n; i++)         
                System.out.print(a[i][n] / a[i][i] +" ");     
        }
    }
      
    // To check whether infinite solutions 
    // exists or no solution exists
    static int CheckConsistency(float a[][], int n, int flag)
    {
        int i, j;
        float sum;
          
        // flag == 2 for infinite solution
        // flag == 3 for No solution
        flag = 3;
        for (i = 0; i < n; i++) 
        {
            sum = 0;
            for (j = 0; j < n; j++)     
                sum = sum + a[i][j];
            if (sum == a[i][j]) 
                flag = 2;     
        }
        return flag;
    }
      
    // Driver code
    public static void main(String[] args) 
    {
        float a[][] = {{ 0, 2, 1, 4 }, 
                        { 1, 1, 2, 6 }, 
                        { 2, 1, 1, 7 }};
                          
        // Order of Matrix(n)
        int n = 3, flag = 0;
          
        // Performing Matrix transformation
        flag = PerformOperation(a, n);
          
        if (flag == 1)     
            flag = CheckConsistency(a, n, flag); 
      
        // Printing Final Matrix
        System.out.println("Final Augumented Matrix is : ");
        PrintMatrix(a, n);
        System.out.println("");
          
        // Printing Solutions(if exist)
        PrintResult(a, n, flag);
    }
    }
      
    /* This code contributed by PrinciRaj1992 */

    C#




    // C# Implementation for Gauss-Jordan
    // Elimination Method
    using System;
    using System.Collections.Generic;
      
    class GFG
    {
    static int M = 10;
      
    // Function to print the matrix
    static void PrintMatrix(float [,]a, int n)
    {
        for (int i = 0; i < n; i++) 
        {
            for (int j = 0; j <= n; j++) 
                Console.Write(a[i, j] + " ");
            Console.WriteLine();
        }
    }
      
    // function to reduce matrix to reduced
    // row echelon form.
    static int PerformOperation(float [,]a, int n)
    {
        int i, j, k = 0, c, flag = 0;
          
        // Performing elementary operations
        for (i = 0; i < n; i++)
        {
            if (a[i, i] == 0) 
            {
                c = 1;
                while ((i + c) < n && a[i + c, i] == 0) 
                    c++;         
                if ((i + c) == n) 
                {
                    flag = 1;
                    break;
                }
                for (j = i, k = 0; k <= n; k++) 
                {
                    float temp = a[j, k];
                    a[j, k] = a[j + c, k];
                    a[j + c, k] = temp;
                }
            }
      
            for (j = 0; j < n; j++) 
            {
                  
                // Excluding all i == j
                if (i != j) 
                {
                      
                    // Converting Matrix to reduced row
                    // echelon form(diagonal matrix)
                    float p = a[j, i] / a[i, i];
      
                    for (k = 0; k <= n; k++)                 
                        a[j, k] = a[j, k] - (a[i, k]) * p;             
                }
            }
        }
        return flag;
    }
      
    // Function to print the desired result 
    // if unique solutions exists, otherwise 
    // prints no solution or infinite solutions 
    // depending upon the input given.
    static void PrintResult(float [,]a,
                            int n, int flag)
    {
        Console.Write("Result is : ");
      
        if (flag == 2)     
        Console.WriteLine("Infinite Solutions Exists"); 
        else if (flag == 3)     
        Console.WriteLine("No Solution Exists");
          
        // Printing the solution by dividing 
        // constants by their respective
        // diagonal elements
        else 
        {
            for (int i = 0; i < n; i++)         
                Console.Write(a[i, n] / a[i, i] + " ");     
        }
    }
      
    // To check whether infinite solutions 
    // exists or no solution exists
    static int CheckConsistency(float [,]a, 
                                int n, int flag)
    {
        int i, j;
        float sum;
          
        // flag == 2 for infinite solution
        // flag == 3 for No solution
        flag = 3;
        for (i = 0; i < n; i++) 
        {
            sum = 0;
            for (j = 0; j < n; j++)     
                sum = sum + a[i, j];
            if (sum == a[i, j]) 
                flag = 2;     
        }
        return flag;
    }
      
    // Driver code
    public static void Main(String[] args) 
    {
        float [,]a = {{ 0, 2, 1, 4 }, 
                      { 1, 1, 2, 6 }, 
                      { 2, 1, 1, 7 }};
                          
        // Order of Matrix(n)
        int n = 3, flag = 0;
          
        // Performing Matrix transformation
        flag = PerformOperation(a, n);
          
        if (flag == 1)     
            flag = CheckConsistency(a, n, flag); 
      
        // Printing Final Matrix
        Console.WriteLine("Final Augumented Matrix is : ");
        PrintMatrix(a, n);
        Console.WriteLine("");
          
        // Printing Solutions(if exist)
        PrintResult(a, n, flag);
    }
    }
      
    // This code is contributed by 29AjayKumar
    Output:
    Final Augumented Matrix is : 
    1 0 0 2.2 
    0 2 0 2.8 
    0 0 -2.5 -3 
    
    Result is : 2.2 1.4 1.2
    

    Applications :

    • Solving System of Linear Equations : Gauss-Jordan Elimination Method can be used for finding the solution of a systems of linear equations which is applied throughout the mathematics.
    • Finding Determinant : The Gaussian Elimination can be applied to a square matrix in order to find determinant of the matrix.
    • Finding Inverse of Matrix : The Gauss-Jordan Elimination method can be used in determining the inverse of a square matrix.
    • Finding Ranks and Bases : Using reduced row echelon form, the ranks as well as bases of square matrices can be computed by Gaussian elimination method.

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