Martin's Rule

  • 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


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  • Q1

    Let a, b, c be positive real numbers. The following system of equations in x, y and z

    x2a2+y2b2-z2c2=1, x2a2-y2b2+z2c2=1, -x2a2+y2b2+z2c2=1 has


    unique solution.


    Let x2a2=X, y2b2=Y, z2c2=Z,

    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.

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  • Q2

    The system of linear equations
    x + y + z = 2
    2x + yz = 3
    3x + 2y + kz = 4
    has a unique solution if


    k 0


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  • Q3

    Using Martin’s rule, solve the following system of equations:
    2x +5y = 6
    5x +3y = 8


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  • Q4

    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


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  • Q5


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