Question

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 \( \beta = \gamma \neq 0 \). 
(ii) \( \alpha = \beta = \gamma = 0 \). 
(iii) \( \beta = \gamma = 0 \) and \( \alpha \neq 0 \). 
(iv) \( \alpha = \beta = \gamma \neq 0 \).
 

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

Hint:

To determine the rank of a matrix, focus on the number of non-zero rows in its row echelon form. If the number of non-zero rows is three or more, the rank is at least three.

Answer 1

The Correct Option is A

We are given the matrix \( A \) and need to determine the rank of \( A \) for different values of \( \alpha, \beta, \) and \( \gamma \).

\[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix} \] Step 1: Analyzing the matrix for different combinations.

Case (i): \( \alpha = 0 \) and \( \beta = \gamma \neq 0 \).
In this case, the matrix becomes:
\[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & 0 & \beta \\ 0 & 0 & 0 & \beta \end{bmatrix} \] Since \( \beta \neq 0 \), there are three non-zero rows, and hence the rank is 3. This combination satisfies the condition.

Case (ii): \( \alpha = \beta = \gamma = 0 \).
In this case, the matrix becomes:
\[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \] This matrix has two non-zero rows, and hence the rank is 2. This combination does not satisfy the condition of rank at least 3.

Case (iii): \( \beta = \gamma = 0 \) and \( \alpha \neq 0 \).
In this case, the matrix becomes:
\[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \] This matrix has three non-zero rows, and hence the rank is 3. This combination satisfies the condition.

Case (iv): \( \alpha = \beta = \gamma \neq 0 \).
In this case, the matrix becomes:
\[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix} \] Since \( \alpha, \beta, \gamma \neq 0 \), there are four non-zero rows, and hence the rank is 4. This combination satisfies the condition.