Using Matrices

In this article we shall study about one of the methods used to solve system of linear equations using matrices. Before we study about the method let us first see few definitions

Consider the following system of simultaneous non – homogeneous linear equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂

Expressing the above equations in matrices, we get

These equations can be represented as a matrix equation as AX = D, where







Here A is called the coefficient matrix.X is called the variable matrix.
D is called the constant matrix.

Augmented Matrices

The coefficient matrix augmented with constant column matrix, is called the augmented matrix, generally denoted by [AD]. Hence the augmented matrix of the above system of simultaneous linear equations is 

Sub Matrices

A matrix obtained by deleting some rows or columns (or both) of a matrix is called a sub matrix.

Definition (Rank of a matrix)

Let A be a non – zero matrices. The rank of A is defined as the maximum of the orders of the non – singular square matrices of A. The rank of a null matrix is defined as zero. The rank of A is denoted by rank (A).

It is to be noted that :-

If A is a non zero matrix of order 3 then rank of A is
(i)    1 if every 2 x 2 sub matrices is singular
(ii)    2 if A is singular and atleast one of its 2 x 2 sub matrices is non – singular.
(iii)    3 if A is non – singular.

Consistent and Inconsistent systems

A system of linear equations is said to be
(i)    Consistent if it has a solution
(ii)    Inconsistent if it has no solution.

A system of three simultaneous equations in three unknowns whose matrix form is AX = D has
(i)    A unique solution if rank (A) = rank ([AD]) = 3
(ii)    Infinitely many solutions if rank (A) = rank([AD]) < 3
(iii)    No solution if rank (A) is not equal to rank ([AD])

It is to be noted that the system is consistent if and only if rank (A) = rank ([AD])

The different ways of solving non homogenous systems of equations are
(i)    Cramer’s Rule
(ii)    Matrix inversion method
(iii)    Gauss – Jordan method

In Gauss – Jordan method we try to transform the augmented matrix by using elementary row transformations. So that the solution is completely visible that is x = α, y = β and z = γ. We may get infinitely many solutions or no solution also.

For solving a system of three equations in three unknowns by Gauss – Jordan method, elementary row operations are performed on the augmented matrix as indicated below.

(i)    Transform the first element of 1^st row and 1^st column to 1 and transform the other non zero elements if any in of 1^st row and 1^st column to zero.
(ii)    Transform the second element of 2^nd row and 2^nd column to 1 and transform the other non zero elements if any in of 2^nd row and 2^nd column to zero.
(iii)    Transform the third element of 3^rd row and 3^rd column to 1 and transform the other non zero elements if any in of 3^rd row and 3^rd column to zero. 

Matrices Calculator
We shall study about Matrices Calculator in some other article.