\(A = \begin{bmatrix}0 & \alpha & \beta\\-\alpha & 0 & \gamma\\-\beta & -\gamma & 0\end{bmatrix}\) is a
(A) square matrix
(B) diagonal matrix
(C) symmetric matrix
(D) skew-symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (D) only
A has 3 rows and 3 columns. Its transpose is
\( \displaystyle A^{T} = \begin{bmatrix}0 & -\alpha & -\beta\\\alpha & 0 & -\gamma\\\beta & \gamma & 0\end{bmatrix} \)
\(A^{T} = -A\)
So A is a 3 × 3 matrix with \(A^{T} = -A\), and its off-diagonal entries α, β, γ are in general non-zero.
Checking the statements
(A) square matrix — Correct. It has 3 rows and 3 columns.
(B) diagonal matrix — Incorrect. The entries α, β, γ off the diagonal need not be zero.
(C) symmetric matrix — Incorrect. \(a_{12} = \alpha\) but \(a_{21} = -\alpha\), so \(A^{T} \ne A\).
(D) skew-symmetric matrix — Correct. \(A^{T} = -A\), with zero diagonal.