GATE ME 2024 Set 1 — Question 14
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Engineering Mathematics → Linear Algebra → Consistency of Linear Systems
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Question
Consider the system of linear equations The values of and such that there exists a non-trivial null space and the system admits infinite solutions are
Correct answer
(D) a = 4, b = 14
Solution
For a non-trivial null space, the determinant of the coefficient matrix must be zero ()..
.Now, for infinite solutions, the rank of the augmented matrix must equal the rank of , and both must be less than the number of variables (3).
With , the augmented matrix is:Perform row operations:
:
:
: For infinite solutions, the last row must be all zeros, so .
Thus, and .
.Now, for infinite solutions, the rank of the augmented matrix must equal the rank of , and both must be less than the number of variables (3).
With , the augmented matrix is:Perform row operations:
:
:
: For infinite solutions, the last row must be all zeros, so .
Thus, and .
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