Question

The Matrix \(M = \begin{bmatrix}        0 & 1 & -1           \\[0.3em]        -1  & 0           &1 \\[0.3em]       1          & -1 & 0     \end{bmatrix}\)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:

• A. (B), (D) Only
• B. (A), (B) Only
• C. (D), (E) Only
• D. (C), (D) Only

Answer 1

The Correct Option is A

The matrix \(M = \begin{bmatrix}0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}\) needs to be analyzed using these properties:

1. Square Matrix: A matrix is square if the number of rows equals the number of columns. Here, \(M\) has 3 rows and 3 columns, so it is a square matrix.
2. Symmetric Matrix: A matrix is symmetric if \(M = M^T\) (transpose). Checking \(M^T = \begin{bmatrix}0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix}\), we see \(M \neq M^T\), so it is not symmetric.
3. Skew-Symmetric Matrix: A matrix is skew-symmetric if \(M = -M^T\). In this case, \(-M^T = \begin{bmatrix}0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}\), and since \(M = -M^T\), it is skew-symmetric.
4. Diagonal Matrix: All off-diagonal elements should be zero. Since \(M\) has non-zero off-diagonal elements, it is not a diagonal matrix.
5. Scalar Matrix: A special diagonal matrix where all diagonal elements are equal. Since \(M\) is not diagonal, it cannot be scalar.

Thus, the correct answer is (B), (D) Only as it is both square and skew-symmetric.