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

Answer: (A)

(A) and (D) only

Answer 2

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 $A^T \neq A$.