Question

3. Consider the following system of linear equations: 2x + 2y + 2kz = 2 kx + ky+z=1 2x + 3y + 7z = 4 (i) Turn the system into row echelon form. (ii) Determine which values of k give (i) a unique solution (ii) infinitely many solutions and (iii) no solutions. Show your working.

Answer 1

3 = 2 + وو * x kw 1 kg ا = 7 + و =17 +3 + في In Matrin form ا = ۹۷ *[ اس کے 6= وة ]ا A - مه lo لايي لك k: کے + w

(a) CAlb] Convert in Row echton form lo 2 2K 2 K k lo 3 7 ५ operate Rg Ro- R 2 2 2K lo 0 -Kiti 0 -K+1 4 ho 3 7 R3 - Roza Ri 2 १ २ 21 - keti -2K+7 -K+1 o 2 R, Rila i K -K²t, KI 0 0 lo ♡ -2K+7

» R3 Rao and Then R, R, Ra 0 3K-7 2 7-2K l- -K+1 0 0 by In Row echlon form b Solution 19) for unique Heto 2 If At If K², to ie K #t! 3K-7 then Rank (: = 3 - 7-2K lokal o ard Rank 0 -1 3k. 7-2K I-K2 - lo = 3 K+I The Rank of Rank of Augemented Matrin is equal to Coefficient Matrin of Syste to Number Variable = 3 tem and is equal

The System is is Consistent ard the Solution Unique li Many Solution, for infinite if K-150 KEL 70 j Ronk 0 3K-7 7-2K l-ka lo o 0 ard Rank 0 3K-7 Par CS = 2 7-2K lo Kd -K +1 The Rank of Augemented Matrin is equal to Coefficient Matrik of System and is less than the number of variables [+ 372) Rank of The System is is Consistent and There is an infinite Number of Solution K=1

liiii for No Solution If K a - 1 I - 1 0 then Rank - 3 2 3K-7 7-2K Inka -K+1 O ard Ronk 3K-7 7-2K 2 0 Ikd The of Rank of Augemented Matrix is . Not equal to Rank of Coefficient Matrine System :: 392) System inconsistent [ Has No Solution) es The k= -1