Finding inverse of a matrix using Gauss - Jordan Method

Given a Matrix, the task is to find the inverse of this Matrix using the Gauss-Jordan method.

What is matrix?

Matrix is an ordered rectangular array of numbers.

Operations that can be performed on a matrix are: Addition, Subtraction, Multiplication or Transpose of matrix etc.

Inverse of a matrix:

Given a square matrix A, which is non-singular (means the Determinant of A is nonzero); Then there exists a matrix $A^-^1$ which is called inverse of matrix A.

The inverse of a matrix is only possible when such properties hold:

The matrix must be a square matrix.
The matrix must be a non-singular matrix and,
There exist an Identity matrix I for which $A A^-^1 = A^-^1 A = I.$

In general, the inverse of n X n matrix A can be found using this simple formula:

where, Adj(A) denotes the adjoint of a matrix and, Det(A) is Determinant of matrix A.

Methods for finding Inverse of Matrix:

Finding the inverse of a 2×2 matrix is a simple task, but for finding the inverse of larger matrix (like 3×3, 4×4, etc) is a tough task, So the following methods can be used:

Elementary Row Operation (Gauss-Jordan Method) (Efficient)
Minors, Cofactors and Ad-jugate Method (Inefficient)

Elementary Row Operation (Gauss – Jordan Method):

Gauss-Jordan Method is a variant of Gaussian elimination in which row reduction operation is performed to find the inverse of a matrix.

Steps to find the inverse of a matrix using Gauss-Jordan method:

In order to find the inverse of the matrix following steps need to be followed:

Form the augmented matrix by the identity matrix.
Perform the row reduction operation on this augmented matrix to generate a row reduced echelon form of the matrix.
The following row operations are performed on augmented matrix when required:
- Interchange any two row.
- Multiply each element of row by a non-zero integer.
- Replace a row by the sum of itself and a constant multiple of another row of the matrix.

Example:

Augmented Matrix is formed as A:B
After applying the Gauss-Jordan elimination method:

Below is the C++ program to find the inverse of a matrix using the Gauss-Jordan method:

// C++ program to find the inverse of Matrix. 
  
#include <iostream> 
#include <vector> 
using namespace std; 
  
// Function to Print matrix. 
void PrintMatrix(float ar[][20], int n, int m) 
{ 
    for (int i = 0; i < n; i++) { 
        for (int j = 0; j < m; j++) { 
            cout << ar[i][j] << "  "; 
        } 
        printf("\n"); 
    } 
    return; 
} 
  
// Function to Print inverse matrix 
void PrintInverse(float ar[][20], int n, int m) 
{ 
    for (int i = 0; i < n; i++) { 
        for (int j = n; j < m; j++) { 
            printf("%.3f  ", ar[i][j]); 
        } 
        printf("\n"); 
    } 
    return; 
} 
  
// Function to perform the inverse operation on the matrix. 
void InverseOfMatrix(float matrix[][20], int order) 
{ 
    // Matrix Declaration. 
  
    float temp; 
  
    // PrintMatrix function to print the element 
    // of the matrix. 
    printf("=== Matrix ===\n"); 
    PrintMatrix(matrix, order, order); 
  
    // Create the augmented matrix 
    // Add the identity matrix 
    // of order at the end of original matrix. 
    for (int i = 0; i < order; i++) { 
  
        for (int j = 0; j < 2 * order; j++) { 
  
            // Add '1' at the diagonal places of 
            // the matrix to create a identity matirx 
            if (j == (i + order)) 
                matrix[i][j] = 1; 
        } 
    } 
  
    // Interchange the row of matrix, 
    // interchanging of row will start from the last row 
    for (int i = order - 1; i > 0; i--) { 
  
        // Swapping each and every element of the two rows 
        // if (matrix[i - 1][0] < matrix[i][0]) 
        // for (int j = 0; j < 2 * order; j++) { 
        // 
        //        // Swapping of the row, if above 
        //        // condition satisfied. 
        // temp = matrix[i][j]; 
        // matrix[i][j] = matrix[i - 1][j]; 
        // matrix[i - 1][j] = temp; 
        //    } 
  
        // Directly swapping the rows using pointers saves time 
  
        if (matrix[i - 1][0] < matrix[i][0]) { 
            float* temp = matrix[i]; 
            matrix[i] = matrix[i - 1]; 
            matrix[i - 1] = temp; 
        } 
    } 
  
    // Print matrix after interchange operations. 
    printf("\n=== Augmented Matrix ===\n"); 
    PrintMatrix(matrix, order, order * 2); 
  
    // Replace a row by sum of itself and a 
    // constant multiple of another row of the matrix 
    for (int i = 0; i < order; i++) { 
  
        for (int j = 0; j < order; j++) { 
  
            if (j != i) { 
  
                temp = matrix[j][i] / matrix[i][i]; 
                for (int k = 0; k < 2 * order; k++) { 
  
                    matrix[j][k] -= matrix[i][k] * temp; 
                } 
            } 
        } 
    } 
  
    // Multiply each row by a nonzero integer. 
    // Divide row element by the diagonal element 
    for (int i = 0; i < order; i++) { 
  
        temp = matrix[i][i]; 
        for (int j = 0; j < 2 * order; j++) { 
  
            matrix[i][j] = matrix[i][j] / temp; 
        } 
    } 
  
    // print the resultant Inverse matrix. 
    printf("\n=== Inverse Matrix ===\n"); 
    PrintInverse(matrix, order, 2 * order); 
  
    return; 
} 
  
// Driver code 
int main() 
{ 
    int order; 
  
    // Order of the matrix 
    // The matrix must be a square a matrix 
    order = 3; 
    float matrix[20][20] = { { 5, 7, 9 }, 
                             { 4, 3, 8 }, 
                             { 7, 5, 6 }, 
                             { 0 } }; 
  
    // Get the inverse of matrix 
    InverseOfMatrix(matrix, order); 
  
    return 0; 
} 

Output:

=== Matrix ===
5  7  9  
4  3  8  
7  5  6  

=== Augmented Matrix ===
7  5  6  0  0  1  
5  7  9  1  0  0  
4  3  8  0  1  0  

=== Inverse Matrix ===
-0.210  0.029  0.276  
0.305  -0.314  -0.038  
-0.010  0.229  -0.124

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My Personal Notes

Finding inverse of a matrix using Gauss – Jordan Method | Set 2

Inverse of a matrix:

Methods for finding Inverse of Matrix:

Elementary Row Operation (Gauss – Jordan Method):

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