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

For skew-symmetric matrices $A$ and $B$:

$A^T=-A,\; B^T=-B$.

Odd powers remain skew-symmetric:

$(A^{2k+1})^T=-(A^{2k+1})$

Even powers become symmetric:

$(A^{2k})^T=A^{2k}$

Now check each statement:

(A) $A^3+B^5$ → both terms are odd powers → both skew-symmetric → sum is skew-symmetric ✔

(B) $A^{19}$ → power is odd → skew-symmetric ✔

(C) $B^{14}$ → power is even → symmetric ✔

(D) $A^4 + B^5$ → $A^4$ is symmetric, $B^5$ is skew-symmetric → sum is neither symmetric nor skew-symmetric ✘

Hence, NOT true statement: $A^4 + B^5$ is symmetric.