Question

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise state that there is no solution. [1 2 -3 51 701 4.5 0000] A) x1 = 15+ 11x3 x2 = -5- 4x3 x3 is free C) x1 = 5 - 2x2 + 3x3 x2 = -5- 4x3 X3 is free B) x1 = 5 - 2x2 + 3x3 x2 is free x3 is free D) x1 = 15+ 11x3 x2 =-5 - 4x3 X3 = 0 Determine whether the system is consistent 4) x1 - x2 + 4x3 = 15 - 4x1 + 4x2 - 16x3 = 4 *1 + 4x2 + x3 = 0 A) NO B) Yes

Answer 1

The given augmented matrix is 2 - 3 5 à = o 4 -5 Loooo and the co-efficient matrix comesponding augmented matrix is to these A= [1 2 -37 Ā Here we denote the augmented matrix by and the co-efficient matrix by A. from the above two matrices we have pank of à = 2 and rank of A=2 Since rank of Ā = park of A=2 Hence the system consistent and the of System is given by solution

4 + 2dz -3dz = 5 M + 4 23 = -5 or " = 5-22 + 3 % z = -5- 483 Since pank of A=2 and there are three Variables say any and , so one Variable must be say z is free Therefore the solution of the system is given by sy = 5-2R₂ + BA3 d = -5-403 Mz is free. So option (C) is right 27 The given lystem is - ₂ + 40 = 15 - 4x + 4z - 16 Az = 4 4 + 4 + d = 0 Therefore, the augmented matrix of the is given by System
A linear non homogeneous system is consistent if  the rank of the augmented matrix and the rank of the coefficient matrix is same. We use this result to check the consistence of the given systems.