GATE CS 2026 Set 1 — Question 20
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Engineering Mathematics → Linear Algebra → Matrices & Determinants
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Question
Let . Consider an matrix with its elements from . Let the vector be in the null space of .Which of the following options is/are always correct?
Correct answer
(B) Determinant of M is 0; (D) There are at least two non-zero vectors in the null space of M
Solution
Let . Since is in the null space of , we have . This implies that the second column of is a zero vector (since is a linear combination of columns of with coefficients from ).
- Option (A) is False: Since one column is zero, the determinant is 0, not 1.
- Option (B) is True: A matrix with a zero column has a determinant of 0.
- Option (C) is False: The rank is at most , but it is not necessarily 1. For example, if is the zero matrix, rank is 0. If has linearly independent columns, rank is .
- Option (D) is True: The null space contains the vector . Since the field is , the null space is a subspace containing the span of , i.e., . This set contains infinite distinct non-zero vectors (e.g., ). Thus, there are at least two non-zero vectors.
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