Let n > 1. Consider an n n matrix M with its elements from R. Let the vector (0, 1, 0, 0, , 0) Rn be in the null space of M. Which of the following options is/are correct?

GATE 2026 CS (CS1) – Question 20

Engineering Mathematics · Linear Algebra · 1 mark · Multiple select

Let $n > 1$. Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$. Let the vector $(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n$ be in the null space of $M$. Which of the following options is/are correct?

Determinant of $M$ is 1
Determinant of $M$ is 0
Rank of $M$ is 1
There are at least two non-zero vectors in the null space of $M$

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Correct answer: (B) Determinant of $M$ is 0; (D) There are at least two non-zero vectors in the null space of $M$

Explanation

Let $v = (0, 1, 0, \dots, 0)^T$. Since $v \in \text{Null}(M)$ and $v \ne 0$, we have $M v = 0$.
1. Because a non-zero vector exists in the null space, the null space has dimension $\ge 1$. Hence, $M$ is singular (not invertible), which means $\det(M) = 0$. (Option B is true; Option A is false).
2. The rank of $M$ is at most $n - 1$. For $n > 2$, the rank does not have to be 1. (Option C is not necessarily true).
3. Since $\text{Null}(M)$ is a subspace over $\mathbb{R}$, for any scalar $c \in \mathbb{R}$, $c \cdot v \in \text{Null}(M)$. In particular, $v$ and $2v$ are two distinct non-zero vectors in $\text{Null}(M)$ (in fact, there are infinitely many). (Option D is true).

Therefore, options (B) and (D) are correct.