Consider the following statements in respect of symmetric matrices A and B.
I. AB is symmetric
II. `A^(2)+B^(2)` is symmetric
Which of the above statements is/are correct ?

Options

A. A. Only I
B. B. Only II
C. C. Both I and II
D. D. Neither I nor II

Answer

To determine the correctness of the statements regarding symmetric matrices \( A \) and \( B \), we will analyze each statement step by step.

### Step 1: Understanding Symmetric Matrices
A matrix \( A \) is called symmetric if it is equal to its transpose, i.e.,
\[ A^T = A \]
Similarly, for matrix \( B \):
\[ B^T = B \]

### Step 2: Analyzing Statement I: \( AB \) is symmetric
To check if \( AB \) is symmetric, we need to find the transpose of the product \( AB \):
\[ (AB)^T = B^T A^T \]
Using the property of transposes, we know that \( (XY)^T = Y^T X^T \) for any matrices \( X \) and \( Y \).

Since both \( A \) and \( B \) are symmetric, we can substitute:
\[ (AB)^T = B A \]
Now, for \( AB \) to be symmetric, we need:
\[ (AB)^T = AB \]
This means:
\[ BA = AB \]
However, this is not always true for arbitrary matrices \( A \) and \( B \). Therefore, we conclude that:
\[ AB \text{ is not necessarily symmetric.} \]
Thus, **Statement I is incorrect**.

### Step 3: Analyzing Statement II: \( A^2 + B^2 \) is symmetric
To check if \( A^2 + B^2 \) is symmetric, we need to find the transpose of \( A^2 + B^2 \):
\[ (A^2 + B^2)^T = A^2 + B^2 \]
Using the property of transposes, we can write:
\[ (A^2 + B^2)^T = A^2 + B^2 \]
Now, we find the transpose of \( A^2 \) and \( B^2 \):
\[ A^2 = A \cdot A \implies (A^2)^T = (A \cdot A)^T = A^T \cdot A^T = A \cdot A = A^2 \]
Similarly for \( B^2 \):
\[ B^2 = B \cdot B \implies (B^2)^T = (B \cdot B)^T = B^T \cdot B^T = B \cdot B = B^2 \]
Thus, we have:
\[ (A^2 + B^2)^T = A^2 + B^2 \]
This shows that \( A^2 + B^2 \) is symmetric. Therefore, **Statement II is correct**.

### Conclusion
From our analysis:
- Statement I: \( AB \) is symmetric - **Incorrect**
- Statement II: \( A^2 + B^2 \) is symmetric - **Correct**

Thus, the correct answer is that only Statement II is correct.

### Final Answer
Only Statement II is correct.

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