Solving System of Linear Equation by Matrix Inverse

The two or more algebraic equation are called system of equations. The Java program finds solution vector X to a system of three linear equations by matrix inverse method.

a₁₁ x₁ + a₁₂ x₂ + a₁₃ x₃ = b ₁
a₂₁ x₁ + a₂₂ x₂ + a₂₃ x₃ = b ₂
a₃₁ x₁ + a₃₂ x₂ + a₃₃ x₃ = b ₃

The three linear equation is represented by matrix format by separating coefficients and unknown variables

                                  |  a11  a12  a13   |
         Coefficients matrix A  = |  a21  a22  a23   |
                                  |  a31  a32  a33   |

         unknown variable     X = [ x1 x2 x3 ]'
       
         non-homogeneous      b = [ b1 b2 b3 ]' 
          
   

                      
        solve given matrix A and vector b (non-homogeneous vector) 
             find value of vector X    

         Matrix representation 

                 A X = b 
         
         multiply both side by A-1

               A-1 A X  =  A-1 b              
         
         Note - A-1 A  = I (Identity matrix)
                
                I X =  A-1 b 

         Note -  I X  = X  

            X =  A-1 b     
   

Algorithm - Solving System of Linear Equation by Matrix Inverse

The following steps are need to do solving system of linear equation by matrix inverse method.

1. Read 3x3 matrix A and 3x1 b vector


2. Find determinant value D of matrix A
• if the determinant value D is zero
• then print matrix A is singular, and has inverse matrix
• else goto step 3


3. Cofactor matrix - finds cofactor matrix from matrix A.
4. Adjoint matrix (adjmat) - finds adjoint matrix by transposing cofactor matrix


5. find A^-1 = adjmat / D , divide each elements of matrix by D (determinant value) scalar operation over adjoint matrix .
6. solution vector X = A^-1 b , multiply A^-1 by the vector b.
7. print solution vector X

Java program - System of Linear Equation by Matrix Inverse

The program Linear-equation has one constructor and two member functions to find a solution vector to system of linear three linear equations.

constructor LinearEquation - accepts two arguments, one is 2D double array having 3x3 elements and other ID double array having 3x1 elements.

1. multiply - This method used to carry out a matrix and a vector multiplication operation i.e. matrix A is multiplied by a vector b .


2. solve - The method to used to solve the system of three equation by matrix inverse method.

 

import java.util.Arrays;

public class LinearEquation extends Matrix {

 double b[];
 public LinearEquation(double A[][],double b[]) {
      super(A);
      this.b=b;
 }
 
 public  double[] multiply(Matrix imat) {
  
    double sol[] =new double[imat.getNrow()];            
    for(int r=0;r<imat.getNrow();r++) 
    {            
       double temp=0;  
       for(int r2=0;r2<b.length;r2++)  
          temp += imat.getElement(r,r2)*b[r2];    

       sol[r]= temp;           
     }  
      return sol;

   }
 
 
    public  double[] solve() {
  
           System.out.println("Input Matrix A 3x3 " );
           System.out.println(this.toString());

         double det=MatrixOpr2.determinant(this);
  
      if ( det==0) {
         System.out.println("Determinant =" + det);
         System.out.println("Matrix is sinular");
         return null;   
      }
  
      System.out.println("Determinant =" + det);
  

       Matrix cfmat=MatrixOpr2.cofactor(this);
       System.out.println("Cofactor matrix");
       System.out.println(cfmat);
  
       Matrix adjmat=cfmat.transpose();
       System.out.println("Adjoint matrix");
       System.out.println(adjmat);
  
       Matrix imat= Scalar.divide(adjmat, det);
  
       System.out.println("Inverse matrix");
       System.out.println(imat);
  
  
       double sol[]=this.multiply(imat);    
       return sol;
  }

 
 public static void main(String[] args) {
  
  double A[][]= { {4,5,-2},{7,-1,2},{3,1,4}};
  double b[]= {-14,42,28};
  
  LinearEquation le=new LinearEquation(A,b);     
  double solution[]=le.solve();
  
  if ( solution !=null ) {
   System.out.println("Solution of Equation :");
   System.out.println(Arrays.toString(solution));
  }
  

 }

}
 

Java program Linear Equation by Matrix Inverse Output

Input Matrix A 3x3 
4.0  5.0  -2.0  
7.0  -1.0  2.0  
3.0  1.0  4.0  

Determinant of Matrix A =-154.0

Cofactor matrix of A
-6.0  -22.0  10.0  
-22.0  22.0  11.0  
8.0  -22.0  -39.0  

Adjoint matrix of A
-6.0  -22.0  8.0  
-22.0  22.0  -22.0  
10.0  11.0  -39.0  

Inverse matrix of A
0.03896103896103896  0.14285714285714285  -0.05194805194805195  
0.14285714285714285  -0.14285714285714285  0.14285714285714285  
-0.06493506493506493  -0.07142857142857142  0.2532467532467532  

Solution of Equation 
[4.0, -4.0, 4.999999999999999]