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Definition: A system of equations in which all the unknowns occur in the first degree alone is, Session 6, Linear System of Equations, Linear System of Equations:, called a system of linear equations., a11x1+ a12xz +a13%3 + ** ain*n, (1), a11x1+a12x2+ a13x3+ *** ain*n = b2, amix1 + am2*2+ am3%3+ amn*n = bm, If bi = b = bz - bm = 0 in (1), then the system (1) is called Homogeneous system of linear, equations. If not all b,,b2 b3, bm are zero then the system (1) is called non-Homogeneous, system of linear equations. A set of all values x1,x2 x3,., In that satisfies all the equations of, the system is called a solution of the system., %3D, Consistent: If the system of linear equations possesses a solution then it is said to be, consistent., Inconsistent: If the system of linear equations has no solution then it is said to be inconsistent., Note: The system of equation (1) can be represented in the Matrix form as AX = B, where A is, called coefficient Matrix, X is called Matrix of Unknowns and B is called Matrix of constants., Homogeneous System of Equations:, A Homogeneous system of equations can be represented in the Matrix form as AX-0,, where A Is called coefficient Matrix, X is called Matrix of Unknowns, O is a zero matrix is, and it is always consistent., It can be observe that x1 = 0, x2, = 0 .. Xn 0 satisfy all the equations of the system, AX-0. Thus x1 x2%3D, %3D, . Xn 0 Is a solution and Is called the trivial solution (or the, zero solutlon)., A solution that is not trivial is called a non-trivlal solution,, In a non-trivlal solutlon, atleast one of x,X2, .. Xn will be different from zero.
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Matrices, Sem 1: Mathematics, Condition for the existence of non-trivial solution:, Theorem: The homogeneous system of m equations AX = 0 in n unknowns has a non-trivial, solution if and only if the rank of the matrix A is less than n. i.e., p(A) = r < n. Further if, Er<n then the system possesses (n-r) linearly independent solutions., Working rule for finding the solution of the homogeneous system AX = 0:, 1) Write the given system of equations in matrix form AX-D0, 2), Reduce the coefficient matrix A to row-reduced echelon form and hence find the rank of, A. Let p(A) r., 3) 2) 1f p(A)=r=n,then the system has trivial solution or zero solution., 4) If. p(A) =r <n, then the system has a non-trivial solution. In this case write the, equivalent system of equations associated with the echelon form obtained in step 1., The equivalent system will have r equations. Choose (n-r) parameters.in terms of which, write the complete solution of the system., Problems:, Solve the following systems of equations:, 1) 2x-3y +z = 0, x+ 2y – 3z = 0, 4x – y – 2z = 0., Solution: Given 2x-3y +z = 0, x + 2y – 3z = 0, 4x – y - 2z 0., The above system can be written in matrix form AX =0, where A=, and X y, -1, Reduce matrix A In to echelon form using the transformations., 2., Consider A, by applylng R1 R2, by applying R2 R2-2R, and RyR3-4R, by applying R2-and R37R1-982
