If A and B are skew-symmetric matrices, then which of the following i…

The correct answer is Option (4) → \(A^6 + B^7\) is symmetric

Since \(A^T = -A\), for any power n:

    \((A^n)^T = (A^T)^n = (-A)^n = (-1)^n A^n\)

So odd powers of a skew-symmetric matrix are skew-symmetric, and even powers are symmetric. The same holds for B.

Option (1) \(A^5 + B^7\) is skew-symmetric — Correct. Both terms are odd powers, so both are skew-symmetric, and so is their sum.

Option (2) \(A^{21}\) is skew-symmetric — Correct. 21 is odd.

Option (3) \(B^{18}\) is symmetric — Correct. 18 is even.

Option (4) \(A^6 + B^7\) is symmetric — Incorrect. \(A^6\) is symmetric and \(B^7\) is skew-symmetric, so

    \((A^6 + B^7)^T = A^6 - B^7\)

This equals \(A^6 + B^7\) only when \(B^7 = O\), so the statement is not true in general.