GATE CE 2025 Set 1 — Question 12
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Engineering Mathematics → Linear Algebra → Consistency of Linear Systems
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Question
Let and . For to be solvable, which one of the following options is the correct condition on , and :
Correct answer
(B) 3b₁ + b₂ + 2b₃ = 0
Solution
For the system to be solvable, the vector must lie in the column space of . This is equivalent to saying that the augmented matrix must have the same rank as .
Performing row operations on the augmented matrix:For the system to be consistent, the last row must be zero:Multiplying by 2:
Performing row operations on the augmented matrix:For the system to be consistent, the last row must be zero:Multiplying by 2:
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