Let $$A, B, C$$ be $$3 \times 3$$ matrices such that $$A$$ is symmetric and $$B$$ and $$C$$ are skew-symmetric. Consider the statements
(S1) $$A^{13}B^{26} - B^{26}A^{13}$$ is symmetric
(S2) $$A^{26}C^{13} - C^{13}A^{26}$$ is symmetric
Then,

Let $$A$$ be symmetric ($$A^T = A$$), and $$B, C$$ be skew-symmetric ($$B^T = -B$$, $$C^T = -C$$).

Properties of powers of symmetric and skew-symmetric matrices.

If $$A$$ is symmetric, then $$A^n$$ is symmetric for all $$n$$: $$(A^n)^T = (A^T)^n = A^n$$.

If $$B$$ is skew-symmetric, then $$B^2$$ is symmetric: $$(B^2)^T = (B^T)^2 = (-B)^2 = B^2$$.

So $$B^{26} = (B^2)^{13}$$ is symmetric.

If $$C$$ is skew-symmetric, then $$C^{13} = C \cdot (C^2)^6$$. Since $$C^2$$ is symmetric, $$(C^{13})^T = ((C^2)^6)^T \cdot C^T = (C^2)^6 \cdot (-C) = -C^{13}$$. So $$C^{13}$$ is skew-symmetric.

Analyze S1: $$A^{13}B^{26} - B^{26}A^{13}$$.

Let $$P = A^{13}$$ (symmetric) and $$Q = B^{26}$$ (symmetric).

$$(PQ - QP)^T = (PQ)^T - (QP)^T = Q^T P^T - P^T Q^T = QP - PQ = -(PQ - QP)$$

So $$A^{13}B^{26} - B^{26}A^{13}$$ is skew-symmetric, not symmetric.

S1 is false.

Analyze S2: $$A^{26}C^{13} - C^{13}A^{26}$$.

Let $$P = A^{26}$$ (symmetric) and $$R = C^{13}$$ (skew-symmetric).

$$(PR - RP)^T = R^T P^T - P^T R^T = (-R)P - P(-R) = -RP + PR = PR - RP$$

So $$A^{26}C^{13} - C^{13}A^{26}$$ is symmetric.

S2 is true.

Conclusion.

Only S2 is true.

The correct answer is Option A: Only S2 is true.