Inverse Matrix by Gauss Jordan Elimination

It finds N x N inverse matrix for a matrix which has NxN elements by Gauss-Jordan elimination method.

Gauss-Jordan elimination method

• A, matrix has N x N elements
• I, Identity matrix has N x N elements
• A|I is a augmented matrix
• Elementary row operation is applied to augmented matrix A|I and it transforms A|I into I|A^-1.

Algorithm Steps for Inverse matrix by Gauss-Jordan Elimination

     
  Read Matrix A 
  form Augmented matrix,  A|I
  
  For i=1  To  N 
  
    For j=1 To  N

   IF  i not-equal j
           apply RowOperation.add(j,i ,-aii/aji) on augmented matrix A|I      
   End   
    End 
 
 End
 
  divide  each row of A|I  by main diagonal elements, aii     

  

Gauss Jordan Inverse - Java programming code

 
public class Gaussjordan {

 Matrix mat;
 
 public Gaussjordan(double A[][]) {
  
    int row=A.length;
    int col=A[0].length;  
    mat = new Matrix(row,col+col);

   for(int r=0;r<row;r++) {
      for(int c=0;c<col;c++) 
          mat.setElement(r, c, A[r][c]);    
     }
  
    for(int r=0;r<row;r++)    
      mat.setElement(r, col+r, 1);
          
 }
 
 public void maxdiagonal() {
    
  for(int r=0;r<mat.getNrow();r++)  
  {
       double mx=0; int mr=r;   
       for (int c=r;c<mat.getNrow();c++){ 
       if (  Math.abs(mat.getElement(c, r)) > mx ) {
           mx = Math.abs(mat.getElement(c, r));
           mr = c;
         }      
      }       
     if ( mr!=r )
       RowOperation.swap(mat, r, mr);           
  }  
 }
 
 public void divbydiag() {
  
   for(int r=0;r<mat.getNrow();r++)  
    {
      double idia = mat.getElement(r, r);    
      for (int c=mat.getNrow();c<mat.getNcol();c++){
        mat.setElement(r, c, mat.getElement(r, c) / idia); 
       }    
       mat.setElement(r, r, mat.getElement(r, r)/idia); 
    }
 }
 
 
 public void diagonalize() {      

     for(int r=0;r<mat.getNrow();r++) {             
        for(int r2=0;r2<mat.getNrow();r2++) {
            if ( r!=r2 ) { 
              double ratio= mat.getElement(r2, r) / mat.getElement(r, r);
              RowOperation.add(mat, r2, r, -ratio);          
             }         
         }      
      }     
  }
  
 public Matrix inverse() {
        
  this.maxdiagonal();  
  this.diagonalize();  
  this.divbydiag();  
  
  Matrix imat = new Matrix(mat.getNrow(),mat.getNrow());  
  for(int r=0;r<mat.getNrow();r++) {
    for(int c=0;c<mat.getNrow();c++) 
         imat.setElement(r, c, mat.getElement(r,mat.getNrow()+c));   
    }
   return imat;
    
 }
 
 public String toString() {
   return mat.toString();
 }
 
 public static void main(String[] args) {
    
  double A[][]={{3,1,2},{2,-1,1},{1,3,-1}};
  
  Gaussjordan gj = new Gaussjordan(A);  
  Matrix imat =gj.inverse();
  System.out.println( "Inverse matrix A" );
  System.out.println( imat.toString() );
  
  
  Matrix mat = new Matrix(A);
  
  Matrix mat2=MatrixOpr.multiply(imat, mat);
  System.out.println( "Identity matrix" );
  System.out.println( mat2.toString() );

 }
}
 

Gauss Jordan Inverse - Java programming output

 matrix A 
3.0  1.0  2.0  
2.0  -1.0  1.0 
1.0  3.0  -1.0 


Augmented matrix A|I 
3.0  1.0  2.0  1.0  0.0  0.0  
2.0  -1.0  1.0  0.0  1.0  0.0  
1.0  3.0  -1.0  0.0  0.0  1.0  


Inverse matrix A
-0.1818181  0.636363  0.27272  
0.27272  -0.45454  0.09090  
0.63636  -0.727272  -0.45454


Prove I= A*A-1
Identity matrix I
1.0    0.0       0.0  
0.0    1.0      -5.551E-17  
0.0  1.110E-16  1.0