System of simultaneous linear equations:

Suppose the subsequent system of n linear equations in n unknowns:

a₁₁ x₁ + a₁₂ x₂ + .........+ a_1n x_n = b₁

a₂₁ x₁ + a₂₂ x₂ + .........+ a_2n x_n = b₂

.           .                                   .     .

.           .                                   .    .

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

That system of equation may be written in matrix form as

 or AX = B

The n x n matrix A is known as the coefficient matrix of the system of linear equations.

Homogeneous and Non-Homogeneous System of Linear Equations:

A system of equations AX = B is known as a homogeneous system if B = O, where O is a null matrix. Otherwise, it is known as a non-homogeneous system of equations.

Solution of a System of Equations:

Suppose the system of equation AX = B

A set of values of the variables x₁, x₂,...,x_n which concurrently satisfy all the equations is known as a solution of the system of equations.

 Consistent System:

If the system of equations has one or more solutions, then it called a consistent system of equations, or else it is an inconsistent system of equations.

Solution of a Non-Homogeneous System of Linear Equations:  

There are two function of solving a non-homogeneous system of simultaneous linear relation.

(i).        Cramer's Rule

(ii).       Matrix Method:

Suppose the equations

Then the system of equations provided by AX = D has a general solution provided by X = A^-1D.

(i).        If A is singular matrix, and (adjA)D = O, then the system of equations shown by AX = D is consistent with infinitely several solutions.

(ii).       If A is singular matrix, and (adjA)D ≠ O, then the system of equation shown by AX = D is incompatible and has no solution.

Solution of Homogeneous System of Linear Equations:

Consider AX = O be a homogeneous system of n linear relation with n unknowns. Now if A is non-singular then the system of equations will have a general solution i.e. trivial solution and if A is a singular then the system of relations will have infinitely several solutions.

Example:    With the help of matrices, solve the relations; 3x + y + 2z = 3, 2x - 3y - z = -3, x + 2y + z = 4.

 Solution:       We may write the provided equations as

                        AX = B                                                ...(1)

                        Where,  

                        Since,  = 3 (-3 + 2 ) -1 (2 + 1) + 2 (4 +3) = -3 -3 + 14 =  8 ≠ 0

                        From (1),   we have  X  =  A^-1B     ...(2)

                        Now,

                        Hence, from (2)

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