System of Simultaneous Linear Equations

Consider the following system of n linear equations in n unknowns:

a₁₁ x₁ + a₁₂ x₂ +.........+ a_1n x_n = d₁

a₂₁ x₁ + a₂₂ x₂ +.........+ a_2n x_n = d₂

.            .                              .   .

.            .                              .   .

a_n1 x₁ + a_n2 x₂ +.........+ a_nn x_n = d_n

This system of equation can be written in the matrix form as

or AX = D.

The n × n matrix A is called the coefficient matrix of the system of linear equations.

Homogeneous and Non-Homogeneous System of Linear Equations

A system of equations AX = D is called a homogeneous system if D = O. Otherwise it is called a non-homogeneous systems of equations.

Solution of a System of Equations

Consider the system of equation AX = D.

A set of values of the variables x₁, x₂, ......, x_n which simultaneously satisfy all the equations is called a solution of the system of equations.

Consistent System

If the system of equations has one or more solutions, then it is said to be a consistent system of equations, otherwise it is an inconsistent system of equations.

Solution of a Non-Homogeneous System of Linear Equations

There are two methods of solving a non-homogeneous system of simultaneous linear equations.

(i)     Cramer's Rule

(ii)    Matrix Method

(i)     Cramer's Rule:

It is discussed under the topic of Determinants.

(ii)    Matrix Method:

Consider the equations

a₁x + b₁y + c₁z = d₁,

a₂x + b₂y + c₂z = d₂,                                                           ...... (i)

a₃x + b₃y + c₃z = d₃.

If A =     X =       and D =

then the equation (i) is equivalent to the matrix equation

A X = D.                                                                            ...... (ii)

Multiplying both sides of (ii) by the inverse matrix A^-1, we get

A^-1 (AX) = A^-1 D => IX = A^-1D         [·.· A^-1 A = I]

=> X = A^-1 D =>                                  ...... (iii)

where A₁, B₁ etc. are the cofactors of a₁, b₂ etc. in the determinant

Δ =       (Δ ≠ 0).

(i)     If A is a non-singular matrix, then the system of equations given by
AX = D has a unique solutions given by X = A^-1 D.

(ii)    If A is a singular matrix, and (adjA)D = O, then the system of equations given by AX = D is consistent, with infinitely many solutions.

(iii)    If A is a singular matrix, and (adjA)D ≠ O, then the system of equation given by AX = D is inconsistent.

Solution of Homogeneous System of Linear Equations:

Let AX = O be a homogeneous system of n linear equation with n unknowns. Now if A is non-singular then the system of equations will have a unique solution i.e. trivial solution and if A is singular then the system of equations will have infinitely many solutions.

Illustration:

If the system of equations x + ay - z = 0, 2x - y + az = 0 and
ax + y + 2z = 0 has a non trivial solution, then find the value of 'a'.

Solution:

        Using C₂ → C₂ - aC₁, C₃ → C₃ + C₁, we get

        A = = 0.

        => (2 + a)(-1 -2a - 1 + a²) = 0

        => (a + 2) (a² - 2a - 2) = 0

        => a = -2, a = 1 + √3.

Illustration:

Find the value of 'k' for which the system of equations (k + 1)
x + 8y = 4k, kx + (k + 3)y = 3k -1 has no solution.

Solution:

        For the system of equations to have no solution, we must have

        (k+1)/k = 8/(k+3) ≠ 4k/(3k-1)

        => (k + 1) (k + 3) = 8k and 8 (3k - 1) ≠ 4k (k + 3)

        => k² - 4k + 3 = 0 => k = 1, 3.

        For  = 1, 8(3k - 1) = 16 and 4k (k + 3) = 16.

        For k = 3, 8(3k - 1) = 64 and 4k (k + 3) = 72.

        .·. for k = 3, 8(3k - 1) ≠ 4k (k + 3).

        .·. k = 3 is the required value of 'k' for no solution.