GATE IN 2025 | Question: 1

Question

A $2 n \times 2 n$ matrix $A=\left[a_{i j}\right]$ has its elements as

$$ a_{i j}=\begin{cases} \beta & \text { if }(i+j) \text { is odd } \\ -\beta & \text { if }(i+j) \text { is even }
\end{cases}$$

where $n$ is any integer greater than $2$ and $\beta$ is any non-zero real number. The rank of $A$ is

• A. $1$
• B. $2$
• C. $n$
• D. $2 n$

Answer 1

1. The elements $a_{ij}$ of the $2n \times 2n$ matrix $A$ depend on whether $(i + j)$ is even or odd.

   • If $(i+j)$ is even: $a_{ij} = -\beta$

   • If $(i+j)$ is odd: $a_{ij} = \beta$

2. Construct a small sample (e.g., $2 \times 2$):

   • $a_{11}$ (even): $-\beta$; $a_{12}$ (odd): $\beta$

   • $a_{21}$ (odd): $\beta$; $a_{22}$ (even): $-\beta$

     $$A = \begin{bmatrix} -\beta & \beta \\ \beta & -\beta \end{bmatrix}$$

   • Row 2 is exactly $-1$ times Row 1 ($R_2 = -R_1$). This pattern continues for any size $2n \times 2n$, where all odd rows are identical and all even rows are identical (and are negatives of the odd rows).

Since all rows are scalar multiples of the first row, there is only one linearly independent row.

 

Answer 2

A.1

Because all rows will be identical can be made 0 by elementary operations.