The matrix \(\begin{bmatrix} 0 & 1 & -1\\ -1 & 0 & 1\\ 1 & -1 & 0\end{bmatrix}\) is :
(A) Symmetric matrix
(B) Square matrix
(C) Diagonal matrix
(D) Skew-symmetric matrix
(E) Scalar matrix
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B), (D) Only
Let A be the given matrix. Its transpose is obtained by writing the rows as columns:
\( A^T = \begin{bmatrix} 0 & -1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0\end{bmatrix} \)
\( = -\begin{bmatrix} 0 & 1 & -1\\ -1 & 0 & 1\\ 1 & -1 & 0\end{bmatrix} = -A \)
So A is a 3 × 3 matrix with \(A^T = -A\), a zero diagonal and non-zero entries off the diagonal.
Checking the statements
(A) Symmetric matrix — Incorrect. \(a_{12} = 1\) but \(a_{21} = -1\), so \(A^T \ne A\).
(B) Square matrix — Correct. It has 3 rows and 3 columns.
(C) Diagonal matrix — Incorrect. Entries off the diagonal, such as \(a_{12} = 1\), are not zero.
(D) Skew-symmetric matrix — Correct. \(A^T = -A\), that is, \(a_{ij} = -a_{ji}\) for all i, j.
(E) Scalar matrix — Incorrect. A scalar matrix must be diagonal, and this matrix is not.