1. Use parity of powers for symmetric/skew-symmetric matrices

Given:

• $A$ is symmetric, so $A^T = A$.
• $B$ and $C$ are skew-symmetric, so $B^T = -B$ and $C^T = -C$.

Now:

• If $A$ is symmetric, then every power of $A$ is symmetric: $$(A^n)^T = (A^T)^n = A^n.$$
• If $B$ is skew-symmetric, then: $$(B^n)^T = (B^T)^n = (-B)^n = (-1)^n B^n.$$ So:
  • $B^{26}$ is symmetric, since $26$ is even.
• Similarly, $C^{13}$ is skew-symmetric, since $13$ is odd.

Also:

• $A^{13}$ and $A^{26}$ are symmetric.

2. Check Statement S1

Statement S1 is:

Let

Since both $A^{13}$ and $B^{26}$ are symmetric, compute transpose:

Thus $X$ is skew-symmetric, not symmetric in general.

So S1 is false.

3. Check Statement S2

Statement S2 is:

Let

Here:

• $A^{26}$ is symmetric,
• $C^{13}$ is skew-symmetric.

Now take transpose:

Thus $Y$ is symmetric.

So S2 is true.

4. Conclusion

• S1 is false
• S2 is true

Therefore, the correct option is:

5. Comparison with stored answer

Stored correct answer: A

My derived answer: A

So they agree.