Question

Given the matrix: \[A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix}\]

 If rank(A) is at least 3, then what are the possible values of \( \alpha, \beta, \gamma \)?

• A. \( \alpha \neq 0 \), \( \gamma \neq 0 \), \( \beta \) is arbitrary
• B. \( \alpha = 0 \), \( \gamma = 0 \), \( \beta \neq 0 \)
• C. \( \alpha = 0 \), \( \gamma = 0 \), \( \beta = 0 \)
• D. \( \alpha \neq 0 \), \( \gamma = 0 \), \( \beta = 0 \)

Hint:

For an upper triangular matrix, rank is simply the number of nonzero diagonal elements.

Answer 1

The Correct Option is A

Understanding Rank in an Upper Triangular Matrix.
For a matrix in upper triangular form, its rank is equal to the number of nonzero diagonal elements. The diagonal elements of \( A \) are: \[ 2, 6, \alpha, \gamma \] Since rank(A) must be at least 3, we require at least three of these elements to be nonzero. - \( 2 \) and \( 6 \) are already nonzero.
- At least one of \( \alpha \) or \( \gamma \) must be nonzero.
- \( \beta \) does not affect rank, so it can be any value.
Thus, the possible values are: \[ \alpha \neq 0 \quad \text{or} \quad \gamma \neq 0, \quad \beta \text{ is arbitrary.} \]