- Basic application of matrix is to solve a system of linear equations. If a system of linear equations has one or more solutions, it is said to be consistent.
- A square matrix A is invertible if and only if A is a non-singular matrix, i.e., |A| ≠ 0. Inverse of a non-singular matrix A is denoted by A–1. Every non-singular matrix possesses a unique inverse.
- By using inverse of a matrix, we can solve a system of linear equations. This method of solving a system of linear equations is known as Martin’s rule.
- The system must have the same number of equations as the number of variables, i.e., the coefficient matrix of the system must be a square matrix.
- If |A| = 0, then system has either infinitely many solutions or no solution.
- If a system of linear equations has no solution, it is said to be inconsistent.
- A system of equations is said to be consistent if the solution exists either infinite or unique, otherwise it is inconsistent.
- If all the constant terms in a system of linear equations are zero, the system is called homogeneous.
- A homogeneous system of n linear equations in n variables is always consistent.
Keywords: Martin’s Rule, Matrix method for solution of system of linear equations, Solution of homogeneous system of linear equations,Solution of non-homogeneous system of linear equations, Conditions for consistency of a system of linear equations
Let a, b, c be positive real numbers. The following system of equations in x, y and zMarks:1
Now, X+Y–Z =1, X–Y+Z=1, –X+Y+Z = 1
The coefficient matrix is
So, the given system of equations has a unique solution.
The system of linear equations
x + y + z = 2
2x + y – z = 3
3x + 2y + kz = 4
has a unique solution ifMarks:1
Using Martin’s rule, solve the following system of equations:
2x +5y = 6
5x +3y = 8Marks:2
Write the given system of equations in the form of single matrix equation. Also, find the value of a for which the system of equations has infinitely many solutions.
x + y + z=0