GATE CE 2023 Set 2 — Question 37
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Engineering Mathematics → Linear Algebra → Rank of a Matrix
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Question
Two vectors and belong to the null space of a matrix of rank 2. Which one of the following vectors also belongs to the null space?
Correct answer
(A) [1 1 -1 1]^T
Solution
The null space of a matrix is a vector space. For a matrix with rank 2, the dimension of the null space (nullity) is calculated as:Since the two given vectors and are linearly independent (one is not a scalar multiple of the other), they form a basis for the null space. Any other vector in the null space must be a linear combination of these two vectors, i.e., .Let's test the options:
.
Since this is a linear combination of the basis vectors, it belongs to the null space.
- Option (A): .
.
Since this is a linear combination of the basis vectors, it belongs to the null space.
- Option (B): is not a linear combination of and .
- Option (C): . Setting to match the second component gives for the first component, but the third component would be .
- Option (D): . Setting and gives , which does not match the fourth component.
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