Question

If a matrix A is both symmetric and skew-symmetric, then A is:

• A. Diagonal matrix
• B. Zero matrix
• C. Non-singular matrix
• D. Scalar matrix

Hint:

A matrix that is both symmetric and skew-symmetric must be a zero matrix. This is because no non-zero matrix can satisfy both conditions simultaneously.

Answer 1

The Correct Option is B

A matrix \( A \) is said to be symmetric if: \[ A^T = A \] A matrix \( A \) is said to be skew-symmetric if: \[ A^T = -A \] If a matrix \( A \) is both symmetric and skew-symmetric, then we can equate the two conditions: \[ A^T = A \quad \text{and} \quad A^T = -A \] This implies: \[ A = -A \] Thus, \( A \) must be the zero matrix because the only matrix that satisfies \( A = -A \) is the matrix where all elements are zero. Therefore, the matrix \( A \) is a zero matrix.