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GATE 2025 EC – Question 11

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Consider the matrix $A$ below:

$$A=\begin{bmatrix}2&3&4&5\\0&6&7&8\\0&0&\alpha&\beta\\0&0&0&\gamma\end{bmatrix}$$

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

(i) $\alpha=0$ and $\beta=\gamma\ne0$.
(ii) $\alpha=\beta=\gamma=0$.
(iii) $\beta=\gamma=0$ and $\alpha\ne0$.
(iv) $\alpha=\beta=\gamma\ne0$.

  1. Only (i), (iii), and (iv)
  2. Only (iv)
  3. Only (ii)
  4. Only (i) and (iii)

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Show answer and explanation

Correct answer: (A) Only (i), (iii), and (iv)

Explanation

The first two rows are always independent (diagonal entries 2 and 6). (i) Row 3 is $(0,0,0,\beta)$ and row 4 is $(0,0,0,\gamma)$, both non-zero multiples of $(0,0,0,1)$, so rank is 3. (ii) Rows 3 and 4 are zero, so rank is 2. (iii) Row 3 is $(0,0,\alpha,0)\ne0$ and row 4 is zero, so rank is 3. (iv) All diagonal entries are non-zero, so rank is 4. Rank is at least 3 for (i), (iii) and (iv).