Matrix Determinant by Upper Triangular Matrix

Determinant, a properties of matrix determines the matrix is singular or not. Upper triangular 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 upper triangular matrix U by elementary row operation and then multiplication of main diagonal elements is called determinant of the matrix A.

• Read matrix A
• Convert matrix A into U by applying Row operation
• Read Main Diagonal elements from U
• Determinant = product of Main Diagonal elements

Algorithm steps for Determinant by Upper Triangular 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=i+1 To  N   
        RowOperation.add(j,i ,-aii/aji)       
     end   

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

Java program for Determinant by Upper Triangular 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 UpperTriangular() {      
 for(int r=0;r<mat.getNrow();r++) {             
     for(int r2=r+1;r2<mat.getNrow();r2++) {         
       double ratio= mat.getElement(r2, r) / mat.getElement(r, r);
       RowOperation.add(mat, r2, r, -ratio);                                   
    }      
  }     
 }
   
  public double detbyU() {  
  System.out.println( " matrix M " );
  System.out.println( this.toString() );
  
  this.maxdiagonal();
  
  this.UpperTriangular();
  System.out.println( "Upper Trianguar matrix " );
  System.out.println( this.toString() );
      
  double D=mat.getElement(0,0);    
  for(int r=1;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 du =gj.detbyU();
  System.out.println( "Determinant of  matrix M ="+ du);
    
  
  double M3[][]={{4, 5,-2},{7,-1,2},{3,1,4}} ;  
  Determinant dm = new Determinant(M3);
    
  double du3 =dm.detbyU();
  System.out.println( "Determinant of  matrix M ="+ du3);
          
 }
}

Java program output for matrix 2x2 and 3x3 Determinant by Upper Triangular Matrix

matrix M 2x2 
4.0  6.0  
3.0  8.0  

Upper Triangular matrix 
4.0  6.0  
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  

Upper Triangular matrix 
7.0  -1.0  2.0  
0.0  5.5714  -3.1428  
0.0  0.0  3.94871  

Determinant of  matrix M =154.0