Question

If \( A \) and \( B \) are two skew-symmetric matrices, then \( AB + BA \) is:

• A. A skew-symmetric matrix
• B. A symmetric matrix
• C. A null matrix
• D. An identity matrix

Hint:

For skew-symmetric matrices, \( AB + BA \) always results in a symmetric matrix.

Answer 1

The Correct Option is B

If \( A \) and \( B \) are skew-symmetric, then: \[ A^T = -A, \quad B^T = -B. \] For the sum \( AB + BA \), we calculate the transpose: \[ (AB + BA)^T = B^T A^T + A^T B^T = (-B)(-A) + (-A)(-B) = AB + BA. \] Since the transpose equals the matrix itself, \( AB + BA \) is symmetric.
Final Answer: \( \boxed{ {Symmetric}} \)