GATE ECE 2025 | Question: 1

Question

Consider the matrix $A$ below:
$$A=\left[\begin{array}{llll}
2 & 3 & 4 & 5 \\
0 & 6 & 7 & 8 \\
0 & 0 & \alpha & \beta \\
0 & 0 & 0 & \gamma
\end{array}\right]$$

For which of the following combinations of $\alpha, \beta$, and $\gamma$, is the rank of $A$ at least three?

1. $\alpha=0$ and $\beta=\gamma \neq 0$.
2. $\alpha=\beta=\gamma=0$.
3. $\beta=\gamma=0$ and $\alpha \neq 0$.
4. $\alpha=\beta=\gamma \neq 0$.

• A. Only (i), (iii), and (iv)
• B. Only (iv)
• C. Only (ii)
• D. Only (i) and (iii)

Answer 1

The answer is A because to get the rank 3 we need non zero rows . so to get non zero rows 1,3,4 those are possible cases. that's why the option is A .

Answer 2

for upper and lower triangular matrices, for the rank to atleast 3
no of non zero diagonal entries have to be equal or more than 3, so either alpha or gama have to be non zero or both can be non zero.
so option 1 is correct

Answer 3

A is the answer we have to see the question atleast 3