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

To solve the problem, we must first understand the properties of the given matrix \[ A = \begin{bmatrix} 0 & \alpha & \beta \\ -\alpha & 0 & \gamma \\ -\beta & -\gamma & 0 \end{bmatrix} \]. Let's analyze the matrix according to the options provided: 

Square Matrix: A square matrix is one that has the same number of rows and columns. Matrix \( A \) is \( 3 \times 3 \), thus, it is a square matrix. Therefore, option (A) is correct.

Diagonal Matrix: A diagonal matrix has all off-diagonal elements equal to zero. In matrix \( A \), the off-diagonal elements are \( \alpha, \beta, \gamma, -\alpha, -\beta, -\gamma \), which are not zero unless \(\alpha = \beta = \gamma = 0\). Therefore, matrix \( A \) is not necessarily a diagonal matrix unless these conditions are true, so option (B) is not correct in general.

Symmetric Matrix: A symmetric matrix satisfies \( A = A^T \) (i.e., \( a_{ij} = a_{ji} \) for all i, j). In matrix \( A \), \( a_{12} = \alpha \) but \( a_{21} = -\alpha \), which are not equal, confirming \( A \neq A^T \). Therefore, matrix \( A \) is not symmetric, making option (C) incorrect.

Skew-Symmetric Matrix: A skew-symmetric matrix satisfies \( A = -A^T \) (i.e., \( a_{ij} = -a_{ji} \) 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 \( a_{32} = -\gamma \), which are all consistent with the definition of skew-symmetry. Therefore, matrix \( A \) is skew-symmetric, making option (D) correct.

Given the analysis, the correct options are (A) and (D), corresponding to the answer choice: (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 \(A^T \neq A\).