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 symmetric if \( A^T = A \). It is skew-symmetric if \( A^T = -A \). If a matrix \( A \) is both symmetric and skew-symmetric, then \( A^T = A \) and \( A^T = -A \). This leads to \( A = -A \), implying that \( A \) must be the zero matrix. The only matrix satisfying \( A = -A \) is the matrix with all zero elements. Consequently, \( A \) is a zero matrix.