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.

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.

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.

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 (D_j). The variable associated with the jth column, x_j, equals D_j / 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 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).