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Let's solve the problem provided in the image. You are asked to determine which of the following matrices are symmetric, given that \( A \) and \( B \) are \( n \times n \) symmetric matrices:

### A matrix \( M \) is symmetric if \( M = M^T \), where \( M^T \) is the transpose of matrix \( M \).

1. **\( A^2 - 2A + 5I_n \)**

   - Since \( A \) is symmetric, \( A^2 \) (the product of a symmetric matrix with itself) is also symmetric.
   - Scalar multiples of symmetric matrices are symmetric, so \( -2A \) is symmetric.
   - \( I_n \), the identity matrix, is always symmetric.
   - Adding symmetric matrices results in a symmetric matrix.
   
   Therefore, \( A^2 - 2A + 5I_n \) is symmetric.

2. **\( AB = BA \)**

   - Symmetric matrices commute under multiplication if and only if \( AB = BA \).
   - If \( AB = BA \), then the matrix product \( AB \) will be symmetric.
   
   Therefore, this matrix is symmetric if \( AB = BA \).

3. **\( A^T B \)**

   - \( A^T \) is the transpose of \( A \). Since \( A \) is symmetric, \( A^T = A \).
   - Therefore, \( A^T B = AB \). But without additional information about \( AB = BA \), we cannot conclude that \( AB \) is symmetric.
   
   Therefore, we cannot guarantee this matrix is symmetric.

4. **\( 2A - 3B \)**

   - Scalar multiples of symmetric matrices (like \( 2A \) and \( -3B \)) are symmetric.
   - The sum of two symmetric matrices is also symmetric.
   
   Therefore, \( 2A - 3B \) is symmetric.

5. **\( BAB \)**

   - Transpose \( (BAB)^T = B^T A^T B^T \).
   - Since \( A \) and \( B \) are symmetric, \( A^T = A \) and \( B^T = B \), thus \( (BAB)^T = BAB \).
   
   Therefore, \( BAB \) is symmetric.

### Summary of symmetric matrices:
- \( A^2 - 2A + 5I_n \)
- \( AB = BA \) (if this condition holds)
- \( 2A - 3B \)
- \( BAB \)

Would you like further details on any of these steps? Here are five related questions:

1. How do we prove that the product of two symmetric matrices is symmetric?
2. What are the properties of symmetric matrices under scalar multiplication?
3. Can two symmetric matrices always commute under multiplication?
4. Why does the identity matrix \( I_n \) always remain symmetric?
5. How does matrix transposition affect symmetry in general?

**Tip:** When working with matrix products, it's essential to check whether matrices commute, as this often influences symmetry properties.

Let A and B be two n × n symmetric matrices. Which of the following matrices are symmetric? A² - 2A + 5Iₙ, AB = BA, AᵀB, 2A - 3B, BAB.

Explore which matrix expressions, including A² - 2A + 5Iₙ, AB = BA, and BAB, are symmetric when A and B are symmetric n × n matrices. Learn the key concepts of matrix algebra, matrix transposition, and conditions for symmetry.

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