Question

2. (a) The linear system 0 212 +by+C12 02x + b2y +0.2 23.C + b3y + 232 0 = 0 0 has only the zero solution 0 i. (3 marks) Determine the number of solutions of the following system. Explain. 212 + by + C2 = 3 02.2 + b2y + 0,2 = 7 03.2 + b3y + 232 11 bi ii. (3 marks) Is the set W b linearly independent? b3 Explain. (b) (4 marks) The augmented matrix for a linear system has been reduced by row operations to the given row echelon form. Solve the system. 1 0 8 -5 6 0 1 4 -9 3 0 0 1 12

Answer 1

The lineare system ayat biy +G2 = 0 azxt bzy to z=0 azn & bzy to 220 can wollten be ay b) by a b3 G AI AX bi 9 b2 q bz x= 7 J. Z Criven that the system AX = o has only zoco solution |Alto & A enists. system Tue can re A X= b ) A= a brą az by G Ginen nou homogeneous wetten ar ba a big Int. Al enists, by A-1 (AX) = ato (A-LA) X=alb X = A-lb. system Ax=b has unique solution X=alb CIB) ". At enists given by :14 140) so (A) = 32 dim (col (A)) coloumns of Á are linearely independent Hence b) are linearly 62 bz } independent

-5 6 (ط) 4 2 - Let x = then lo 8-5 0 10 4 -9 (3) یسنا 2 - (2 ze wz = 2 7 + 42-gw=3 I + 8z-5 w z 6 3 From O wa 2-Z Brom J z 3+9w-42 = 3+9 (2-2) - Yz CW-2-2 = 3 + 1 8 - 9 Z-42 Brom > 21 -12 Z a z 6 +5w- 87 = 6 + 5(2-2) -82 = 6 +10 -52-82 z 16 - 132. Herce JAN3 = 16-182 21-132 is the required solution 2-2 where conltat. is arebitarey

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