# 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

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-\frac{{z}^{2}}{{c}^{2}}=1,\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1,-\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1has\phantom{\rule{0ex}{0ex}}$

Marks:1

unique solution.

##### Explanation:

Now, X+Y–Z =1, X–Y+Z=1, –X+Y+Z = 1

The coefficient matrix is

$\mathrm{A}=\left|\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right|$

$\left|A\right|=1\left(-1-1\right)-1\left(1+1\right)+1\left(1-1\right)=-4\ne 0$

So, the given system of equations has a unique solution.

• Q2

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

Marks:1

k 0

##### Explanation:

• Q3

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

Marks:2

• 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.
5x+5y+az=0
x + y + z=0
11x+ay+11z=0

Marks:2