Let $$X = A + B$$ and $$Y = A - B$$.
Taking the transpose of $$X$$:
$$X^T = (A + B)^T = A^T + B^T = A - B = Y$$
Taking the transpose of $$Y$$:
$$Y^T = (A - B)^T = A^T - B^T = A - (-B) = A + B = X$$
Therefore, $$X^T = Y$$ and $$Y^T = X$$.
2. Analyzing Statement S1 ($$P$$)
The matrix $$P$$ is defined as $$P = Y^{2023}X^{2023} - X^{2023}Y^{2023}$$.
To check if it is skew-symmetric, find $$P^T$$:
$$P^T = (Y^{2023}X^{2023})^T - (X^{2023}Y^{2023})^T$$
Using the property $$(MN)^T = N^T M^T$$:
$$P^T = (X^T)^{2023}(Y^T)^{2023} - (Y^T)^{2023}(X^T)^{2023}$$
Substitute $$X^T = Y$$ and $$Y^T = X$$:
$$P^T = Y^{2023}X^{2023} - X^{2023}Y^{2023}$$ $$P^T = P$$
Since $$P^T = P$$, matrix $$P$$ is symmetric, not skew-symmetric.
Result: S1 is False.
3. Analyzing Statement S2 ($$Q$$)
Substitute $$A^4 + B^4 = A$$ into the expression for $$Q$$:
$$Q = (A + B)^T A (A - B) + (A - B)^T A (A + B)$$
Using the transpose relationships $$X^T = Y$$ and $$Y^T = X$$:
$$Q = Y A Y + X A X$$
Now, find $$Q^T$$:
$$Q^T = (Y A Y)^T + (X A X)^T$$ $$Q^T = Y^T A^T Y^T + X^T A^T X^T$$
Substitute $$Y^T = X$$, $$X^T = Y$$, and $$A^T = A$$:
$$Q^T = X A X + Y A Y$$ $$Q^T = Q$$
Since $$Q^T = Q$$, matrix $$Q$$ is symmetric.
Result: S2 is True.
Conclusion
- S1 is False (P is symmetric).
- S2 is True (Q is symmetric).
Correct Option: C (Only S2 is true)
91- 6303239042
