Let `A,B,C` be `3times3` matrices such that `A` is symmetric and `B` and `C` are skew-symmetric. Consider the statements
(`S_1`) `A^(13)B^(26)-B^(26)A^(13)` is symmetric
`(S_(2))A^(26)C^(13)-C^(13)A^(26)` is symmetric Then

Options

A. Only `S_(1)` is true
B. Both `S_(1)` and `S_(2)` are false
C. Both `S_(1)` and `S_(2)` are true `
D. Only `S_(2)` is true

Answer

To solve the problem, we need to analyze the two statements regarding the matrices \( A \), \( B \), and \( C \).

### Step 1: Analyze Statement \( S_1 \)

**Statement \( S_1 \)**: \( A^{13}B^{26} - B^{26}A^{13} \) is symmetric.

1. **Transpose the Expression**:
We need to check if the transpose of \( A^{13}B^{26} - B^{26}A^{13} \) is equal to the original expression.
\[ (A^{13}B^{26} - B^{26}A^{13})^T = (A^{13}B^{26})^T - (B^{26}A^{13})^T \]

2. **Apply the Transpose Property**:
Using the property \( (XY)^T = Y^T X^T \):
\[ (A^{13}B^{26})^T = B^{26^T}A^{13^T} \quad \text{and} \quad (B^{26}A^{13})^T = A^{13^T}B^{26^T} \]

3. **Substituting Transpose of Matrices**:
Since \( A \) is symmetric, \( A^{13^T} = A^{13} \). For skew-symmetric matrices, \( B^{26^T} = -B^{26} \):
\[ (A^{13}B^{26} - B^{26}A^{13})^T = (-B^{26})A^{13} - A^{13}(-B^{26}) = -B^{26}A^{13} + B^{26}A^{13} = 0 \]

4. **Conclusion for \( S_1 \)**:
Since the transpose is equal to the negative of the original expression, \( A^{13}B^{26} - B^{26}A^{13} \) is skew-symmetric, not symmetric. Thus, \( S_1 \) is **false**.

### Step 2: Analyze Statement \( S_2 \)

**Statement \( S_2 \)**: \( A^{26}C^{13} - C^{13}A^{26} \) is symmetric.

1. **Transpose the Expression**:
Similar to \( S_1 \):
\[ (A^{26}C^{13} - C^{13}A^{26})^T = (A^{26}C^{13})^T - (C^{13}A^{26})^T \]

2. **Apply the Transpose Property**:
\[ (A^{26}C^{13})^T = C^{13^T}A^{26^T} \quad \text{and} \quad (C^{13}A^{26})^T = A^{26^T}C^{13^T} \]

3. **Substituting Transpose of Matrices**:
Again, \( A^{26^T} = A^{26} \) and \( C^{13^T} = -C^{13} \):
\[ (A^{26}C^{13} - C^{13}A^{26})^T = (-C^{13})A^{26} - A^{26}(-C^{13}) = -C^{13}A^{26} + C^{13}A^{26} = 0 \]

4. **Conclusion for \( S_2 \)**:
Since the transpose is equal to the negative of the original expression, \( A^{26}C^{13} - C^{13}A^{26} \) is skew-symmetric, not symmetric. Thus, \( S_2 \) is also **false**.

### Final Conclusion

Both statements \( S_1 \) and \( S_2 \) are false.