To address this problem, we first examine the properties of the matrix \[ A = \begin{bmatrix} 0 & \alpha & \beta \\ -\alpha & 0 & \gamma \\ -\beta & -\gamma & 0 \end{bmatrix} \]. We will evaluate it against the provided options:
Square Matrix: A square matrix has an equal number of rows and columns. Matrix \( A \) is a \( 3 \times 3 \) matrix, thus it is square. Option (A) is correct.
Diagonal Matrix: A diagonal matrix has all elements outside the main diagonal equal to zero. Matrix \( A \) has off-diagonal elements \( \alpha, \beta, \gamma, -\alpha, -\beta, -\gamma \), which are not zero unless \(\alpha = \beta = \gamma = 0\). Therefore, \( A \) is not necessarily a diagonal matrix, and option (B) is generally incorrect.
Symmetric Matrix: A symmetric matrix satisfies \( A = A^T \), meaning \( a_{ij} = a_{ji} \) for all \( i, j \). For matrix \( A \), \( a_{12} = \alpha \) and \( a_{21} = -\alpha \). Since \( \alpha eq -\alpha \) unless \(\alpha = 0\), \( A \) is not symmetric. Option (C) is incorrect.
Skew-Symmetric Matrix: A skew-symmetric matrix satisfies \( A = -A^T \), meaning \( a_{ij} = -a_{ji} \) for all \( i, j \). In matrix \( A \), \( a_{12} = \alpha \) and \( a_{21} = -\alpha \); \( a_{13} = \beta \) and \( a_{31} = -\beta \); \( a_{23} = \gamma \) and \( a_{32} = -\gamma \). These relationships adhere to the definition of skew-symmetry. Therefore, matrix \( A \) is skew-symmetric, and option (D) is correct.
Based on this analysis, the correct options are (A) and (D). The answer choice is: (A) and (D) only