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GATE 2025 IN – Question 11

Engineering Mathematics · Linear algebra and matrices · 1 mark · Multiple choice

A $2n\times2n$ matrix $A=[a_{ij}]$ has elements $$a_{ij}=\begin{cases}\beta,&i+j\text{ odd},\\-\beta,&i+j\text{ even}.\end{cases}$$ Here $n$ is any integer greater than 2 and $\beta$ is any nonzero real number. The rank of $A$ is

  1. 1
  2. 2
  3. $n$
  4. $2n$

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Correct answer: (A) 1

Explanation

Every row is either the same alternating vector or its negative, because moving one index reverses the parity of i+j.

The matrix therefore has at most one independent row.

Since β≠0 at least one row is nonzero, giving **rank 1, A**.