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### Problem 3: Prove that if \(A\) and \(B\) are symmetric matrices, then \(AB\) is symmetric if and only if \(AB = BA\).

#### Proof:

1. **Definitions**:
   - A matrix \(A\) is **symmetric** if \(A^T = A\), where \(A^T\) is the transpose of \(A\).
   - A matrix \(B\) is **symmetric** if \(B^T = B\).
   - We need to show that \(AB\) is symmetric if and only if \(AB = BA\).

2. **Condition for symmetry of \(AB\)**:
   - For \(AB\) to be symmetric, \((AB)^T = AB\).

3. **Transpose property**:
   - The transpose of a product of two matrices satisfies \((AB)^T = B^T A^T\).

4. **Use the symmetry of \(A\) and \(B\)**:
   - Since \(A\) and \(B\) are symmetric, we have \(A^T = A\) and \(B^T = B\).
   - Substitute these into the transpose property:
     \[
     (AB)^T = B^T A^T = BA.
     \]

5. **Symmetry of \(AB\)**:
   - For \(AB\) to be symmetric, \((AB)^T = AB\). Using the above, this means:
     \[
     BA = AB.
     \]

6. **Conclusion**:
   - \(AB\) is symmetric if and only if \(AB = BA\).

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### Problem 4: Prove that the diagonal entries of a skew-symmetric matrix are zero.

#### Proof:

1. **Definition of a skew-symmetric matrix**:
   - A matrix \(A\) is skew-symmetric if \(A^T = -A\), where \(A^T\) is the transpose of \(A\).

2. **Property of diagonal entries**:
   - Let \(A = [a_{ij}]\) be a square matrix.
   - The \(i\)-th diagonal entry is \(a_{ii}\).

3. **Transpose condition for diagonal entries**:
   - For \(A^T = -A\), the elements of \(A^T\) are \(a_{ji}\) (entries are transposed

Let A and B be symmetric matrices. Show that AB is symmetric if and only if AB = BA. Show that the diagonal entries of a skew-symmetric matrix are zero.

Explore detailed proofs of key matrix properties: when the product of symmetric matrices is symmetric and why the diagonal entries of a skew-symmetric matrix are zero. Includes mathematical concepts, formulas, and relevant theorems.

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