Question

Given \[ A = \begin{bmatrix} 0 & \alpha & \beta \\ -\alpha & 0 & \gamma \\ -\beta & -\gamma & 0 \end{bmatrix}, \] the matrix $A$ is a:
(A) square matrix
(B) diagonal matrix
(C) symmetric matrix
(D) skew-symmetric matrix .
Choose the correct answer from the options given below:

• A. (A) and (D) only
• B. (A) and (C) only
• C. (A), (B) and (C) only
• D. (A), (B) and (D) only

Answer 1

The Correct Option is A

Matrix \(A\) is a \(3 \times 3\) square matrix. To determine if it is skew-symmetric, we check if \(A^T = -A\). The transpose of \(A\) is:

\[ A^T = \begin{bmatrix} 0 & -\alpha & -\beta \\ \alpha & 0 & -\gamma \\ \beta & \gamma & 0 \end{bmatrix}. \]

Clearly, \(A^T = -A\), so \(A\) is a skew-symmetric matrix.

(B) The matrix is not a diagonal matrix since not all non-diagonal elements are zero.

(C) The matrix is not symmetric since \(A^T \neq A\).