Matrix Determinant by Lower Triangular Matrix

Determinant, a properties of matrix determines the matrix is singular or not. Lower Triangular matrix 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 Lower triangular matrix, L 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 L by applying Row operation or reduction technique
• Read Main Diagonal elements from L
• Determinant = product of Main Diagonal elements

Algorithm steps for Determinant by Lower Triangular Matrix

  Read matrix A
  a -  elements of matrix A
  i,j - position of a element in the matrix

  for i=N-1  To 1 decrement by -1 
  
     for j=i-1 To 1 decrement  by -1   
        RowOperation.add(j,i ,-aii/aji)       
     end   

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

  

Java program for Determinant by Lower 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  Matrix LowerTriangular() {
  
 for(int c=mat.getNrow()-1;c>0;c--) {                  
     for(int c2=c-1;c2>=0;c2--) 
  {  
          double ratio= mat.getElement(c2, c) / mat.getElement(c, c);
          RowOperation.add(mat, c2, c, -ratio);               
        
   }                
       }  
     return mat;
     }
 
 public double detbyL() {
  
 System.out.println( " matrix M " + mat.getNrow() +" x " 
       + mat.getNcol());
 System.out.println( this.toString() );
  
 this.maxdiagonal();
  
 this.LowerTriangular();
 System.out.println( "Lower 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 dl =gj.detbyL();
  System.out.println( "Determinant of  matrix M ="+ dl);
  
  
  double M3[][]={{4, 5,-2},{7,-1,2},{3,1,4}} ;  
  Determinant dm = new Determinant(M3);
  
  
  double dl3 =dm.detbyL();
  System.out.println( "Determinant of  matrix M ="+ dl3);
      
 }

}

Java program output for Determinant by Lower Triangular Matrix

matrix M 2x2
4.0  6.0  
3.0  8.0  

Lower Triangular matrix 
1.75  0.0  
3.0  8.0  

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  

Lower Triangular matrix 
7.0  0.0  0.0  
5.5  5.5  0.0  
3.0  1.0  4.0  

Determinant of  matrix M =154.0