Matrix Determinant by Diagonal Matrix

A matrix diagonal transformation method is preferred over minor or cofactor of matrix method while finding determinant of the matrix's size over 3x3.

The matrix A is converted into Diagonal matrix D by elementary row operation or reduction and then product of main diagonal elements is called determinant of the matrix A.

• Read matrix A
• Convert matrix A into diagonal matrix D by applying Row operation or reduction technique
• Read Main Diagonal elements from D
• Determinant = product of Main Diagonal elements

Algorithm steps for Determinant by Diagonal matrix

  Read matrix A
  a - element of the matrix A
  i,j - position of a element in the matrix  
  
  for i=1  To  N 
  
    for j=1 To  N
 if i not-equal j
           RowOperation.add(j,i ,-aii/aji)       
  end   
    end 

 end
 
  Determinant = a11  x a22 x ... x ann

 

Java program for a matrix Determinant by Diagonal matrix

 
public class Determinant {

 Matrix mat;
 public Determinant(double M[][]) {
      
  int row=M.length;
  int col=M[0].length;
  
  mat = new Matrix(row,col);
  for(int r=0;r<row;r++) {
   for(int c=0;c<col;c++) 
        mat.setElement(r, c, M[r][c]);    
  }
 }
 
 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 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 double detbyD() {
  
  System.out.println( " matrix A " );
  System.out.println( this.toString() );
  
  this.maxdiagonal();
  
  this.diagonalize();
  System.out.println( "Diagonal matrix " );
  System.out.println( this.toString() );
      
  double D=1.0;    
  for(int r=0;r<mat.getNrow();r++) {   
        D= D * mat.getElement(r,r);   
  }
    return D;
    
 }
    
   public String toString() {
   return mat.toString();
 }
 
 public static void main(String[] args) {
  
  
  double M2[][]={{4, 6},{3,8}} ;  
  Determinant gj = new Determinant(M2);  
  
  double dd =gj.detbyD();
  System.out.println( "Determinant of  matrix M ="+ dd);
  
  
  double M3[][]={{4, 5,-2},{7,-1,2},{3,1,4}} ;  
  Determinant dm = new Determinant(M3);
  
  double dd3 =dm.detbyD();
  System.out.println( "Determinant of  matrix M ="+ dd3);
              
 }
}

Java program output for matrix 2x2 and 3x3 Determinant by Diagonal matrix

matrix M 2x2 
4.0  6.0  
3.0  8.0  

Diagonal matrix 
4.0  0.0  
0.0  3.5  

Determinant of  matrix M =14.0

matrix M 3x3 
4.0  5.0  -2.0  
7.0  -1.0  2.0  
3.0  1.0  4.0  

Diagonal matrix 
7.0  0.0  0.0  
0.0  5.57142  0.0  
0.0  0.0  3.94871  

Determinant of  matrix M =154.0