Consider the following system of n linear equations in n unknowns:
a11 x1 + a12 x2 +.........+ a1n xn = d1
a21 x1 + a22 x2 +.........+ a2n xn = d2 . . .
an1 x1 + an2 x2 +.........+ ann xn = dn
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, ......, xn 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.
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.
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.
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