Question

4. Consider the linear system /1 2 T2 -2 1 2 1 3 22-3) 4 :) [4] (a) Determine the value(s) of a, if any, for which this system has precisely one solution [2] (b) Determine the value(s) of a, if any, for which this system has no solutions [2] (c) Determine the value(s) of a, if any, for which this system has infinitely many solutions

Answer 1

Griven question. I a i n AX=D This is of the form Here Aar 21 12- 23 1.2 Now Augmented mabaix is written of ADI 21:2 1 2 02. 3a Here we have three conditions the 1. If rank of A E Bank of AD - order of trotz order = No. of variables = 3 [order of A) then the system has one Solution. 2 of rank of A = rank of AD e order of matrix <3 then the system how Infinitely many solution 3. I rank of A & rank of AD . then the system has in solution

Now to find rank of A and to wing Elementary row operations by converting mabrix li 2 11 27 into Echeleon form. | 2 -2 311 Li 2 azzia ] R₂ R₂-h . ri 21:27 12 m2 311 Lo o 224:az] R2 R2-2B1 2 1 : 2 7 (rank of a matrix is no of: Non Z210 POWST [2ܘ : ܚܐ. ܘ . ] j) in a matrix . a Hey investing it into echelen form for one Solution kle have Rank y A - Bank op A3 order :3 This is possible only when a24 to &a-24 at +2 a $2 Nove This is because if a=+2 then Rank of 4 33 23 il a=2 then lank of AD = 2 which is not eaud NO. cf.no zety to No of variables (no-of-no 17 zero reus) in D

if a=-2 then If a rank of A=2 so for one solution a can be any value Other than $2. 3) If rank of Atrank of AD Then system has - No solution. 7 rankof AD-3 This is possible only if.2--2 but if a=2 then rank of in rank of AD so :: it's not possible to have no Solection, för system having Infinite No. of solutions Bank of Az rank of AD < Nos ay variables This is possible only if a=2,1 m if a=2, Bank of A = rank of (AP) = 243 2 . Bunun (no. Of. nonzero 9 rows)
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