If A and B are skew-symmetric matrices, then which one of the followi…

The correct answer is Option (4) → \(A^4 + B^5\) is symmetric

Since A is skew-symmetric, \(A^T = -A\), so for any positive integer n:

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

So an odd power of a skew-symmetric matrix is skew-symmetric, and an even power is symmetric. The same holds for B.

Option (1) \(A^3 + B^5\) is skew-symmetric — Correct. \((A^3 + B^5)^T = -A^3 - B^5 = -(A^3 + B^5)\).

Option (2) \(A^{19}\) is skew-symmetric — Correct. 19 is odd.

Option (3) \(B^{14}\) is symmetric — Correct. 14 is even.

Option (4) \(A^4 + B^5\) is symmetric — Incorrect. \((A^4 + B^5)^T = A^4 - B^5\), which equals \(A^4 + B^5\) only when \(B^5 = O\). This fails in general: for

    \( \displaystyle B = \begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix},\quad B^2 = -I,\quad B^5 = B \ne O \)

So the statement that is NOT true is Option (4).