Solution of simultaneous equations

Solution of simultaneous equations

Solving Simultaneous Equations Using Matrices

  • Simultaneous equations can also be solved using matrices and matrix operations. The system of equations can be represented in the form AX = B where A is the matrix of coefficients, X is the matrix of variables and B is the matrix of constants.

  • To solve for X (variables), we can multiply both sides of the equation by the inverse of matrix A, provided it exists. The resulting equation would be X = A-1B. This allows us to solve for the variables.

Check for Consistency

  • Not all systems of linear simultaneous equations have solutions. They may be consistent (having one or infinite solutions) or inconsistent (having no solution).

  • A system is inconsistent if the rank of the coefficient matrix A is not equal to the **rank of the augmented matrix [A B]**. In other words, if rank(A)!=rank([A B]), system is inconsistent.

Number of Solutions

  • The number of possible solutions can also be determined using the rank of the matrices:
    • If rank(A)=rank([A B])=n (where n is number of variables), then system is consistent and has unique solution.
    • If rank(A)=rank([A B])<n, then system is consistent and has infinite solutions.
    • If rank(A) ≠ rank([A B]), system is inconsistent and has no solution.

Using Cramer’s Rule

  • For a system of n simultaneous equations to have a unique solution, the determinant of the coefficient matrix ( A ) must not be zero. If A ≠ 0, the system can be solved using Cramer’s rule.
  • Cramer’s rule involves replacing the jth column of A with B to form a new matrix and calculating its determinant (Dj). The variable associated with the jth column, xj, equals Dj / A .
  • Cramer’s rule can be time-consuming for larger matrices. However, it provides a direct method to find solutions without having to calculate matrix inverses.

Gaussian Elimination and Gauss-Jordan Method

  • Gaussian elimination and Gauss-Jordan method are two methods often used to solve systems of simultaneous equations. They transform the system to a simpler one, row by row, until the solutions can be read directly.

  • These methods involve row operations, including swapping rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another row. The goal is to create a upper triangular matrix (Gaussian) or diagonal matrix (Gauss-Jordan) to solve.

  • Gaussian elimination involves more steps because it only reduces the matrix to row echelon form (upper triangular) whereas Gauss-Jordan reduces it to reduced row echelon form (diagonal).