GATE ME 2022 Set 1 — Question 46
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Engineering Mathematics → Linear Algebra → Consistency of Linear Systems
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Question
The system of linear equations in real given byinvolves a real parameter and has infinitely many non-trivial solutions for special value(s) of . Which one or more among the following options is/are non-trivial solution(s) of for such special value(s) of ?
Correct answer
(A) x = 2, y = -2; (B) x = -1, y = 4
Solution
The given system of linear equations is . This can be expanded as:
1)
2) For the system to have infinitely many non-trivial solutions, the determinant of the coefficient matrix must be zero:This gives or .Case 1:
The equations become .
Checking the options:
(A) (Satisfied)
(D) (Not satisfied)Case 2:
The equations become .
Checking the options:
(B) (Satisfied)
(C) (Not satisfied)Thus, the non-trivial solutions are given in options (A) and (B).
1)
2) For the system to have infinitely many non-trivial solutions, the determinant of the coefficient matrix must be zero:This gives or .Case 1:
The equations become .
Checking the options:
(A) (Satisfied)
(D) (Not satisfied)Case 2:
The equations become .
Checking the options:
(B) (Satisfied)
(C) (Not satisfied)Thus, the non-trivial solutions are given in options (A) and (B).
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