Question

Determine a, b E R so that the system 6x1 + (2a - 3)x2 + 4ax3 + 324 = 2 + b 4x1 + (2a - 2)x2 + 2axy + 2x4 = 2 4x1 - 2x2 + (4a + 2)x3 + (2a + 2)x1 = 0 201 - 22 + (2a +3)x3 + 2ax 1 = -2b - 2 has (i) a unique solution, (ii) infinitely many solutions, (iii) no solution

Answer 1

e Given system of equations 6x + (2a-3) x2 + 4 axz + 3x4 = a tb 4x + (20-2) x2 + 2ax3 + 204 4x, - & xa + (4a +2) X3 .it (2a + a) xy = 0 2x - x2 + (2x+3)x3 + Q any = -26-2 This can be written 3 2 + 2a-3 4a lo KO x2 4 ga-2 Qa O & at a hata - 26-2 4 24 Qat3 ko A X = B augmented matria [AB] is 3 atb 6 ga-3 ya 1 > ga-2 2a 1 Qata hat2 o 2a+3 Qa! -2b-2 apply Ra -7 3 Rg 2R1, R₂ -> 3 R3 -2R, R4 - 6 Ry-R, Qa-3 4a 3 2+b 6 o o -Qa O 2-2b -4a 40+6 6a -4-2b 29 + 9 6a-3 -7b-8

apply R₂ -> R3 ta R& - R4 +Ra 1 276 3 2a-3 4a 2-26 O - 1 6b 6a 9 o GA-3 i-6b-9b apply R4 > & Ry - 3 R3 3 a+b чо qa-3 ca 2-gb 1 6a I - 6b O 6 -6a-6! -12 e Casi: If a then case i: If a=-1, for any b then Rank of A=3 (no of nonzero rows of A) Rank of AB = 4 (no of nonzero rows of AB) Rank A & Rank ABG ... The system has no solution

case ii : If -60 - 670, 3 -6a76 b lat-1 , for any then Rank of A = 4 Rank of AB= 4 no.. Variables 4 . Rank A variables = Rank AB = no.org The system has unique solution Case ili If a=- and b=2 then Rank of A= 3 Rank of AB = 3 no es variables=4 i. Rank of A= Rank on AB <no. of variables hence the system has infinite number of solutions, ii) a unique solution is at-1 and for any value of solutions if a = -1 and b=2 il) infinitely many in) no solution ich a=-1 and for any value of b