Mark the following True or False:
1 If the (3, 1)-entry of a matrix A is 5, then the (1, 3)-entry of AT is −5.
2 A and AT have the same main diagonal for every matrix A.
3 If B is symmetric and AT = 3B, then A = 3B.
4 If A and B are symmetric, then kA + mB is symmetric for any scalars
k and m.
5 If A is a square matrix, then A + AT is skew-symmetric.

Let's examine each statement one by one and determine whether they are **True** or **False**:

### 1. **If the (3, 1)-entry of a matrix A is 5, then the (1, 3)-entry of \(A^T\) is −5.**
- The transpose of a matrix \(A\), denoted \(A^T\), involves swapping the rows and columns. If the (3, 1)-entry of \(A\) is 5, then the (1, 3)-entry of \(A^T\) will be **5**, not −5. Transposition doesn't change the sign of an element.
- **Answer: False**

### 2. **A and \(A^T\) have the same main diagonal for every matrix \(A\).**
- The main diagonal entries of a matrix \(A\) are the same as those of its transpose \(A^T\) because the diagonal elements are not affected by transposition (i.e., they remain in the same position).
- **Answer: True**

### 3. **If \(B\) is symmetric and \(A^T = 3B\), then \(A = 3B\).**
- For a symmetric matrix \(B\), \(B^T = B\). However, the condition \(A^T = 3B\) only implies that the transpose of \(A\) is \(3B\), not necessarily that \(A = 3B\). In fact, if \(A^T = 3B\), it does not directly lead to \(A = 3B\) without further conditions on \(A\).
- **Answer: False**

### 4. **If \(A\) and \(B\) are symmetric, then \(kA + mB\) is symmetric for any scalars \(k\) and \(m\).**
- A matrix is symmetric if \(A^T = A\). If \(A\) and \(B\) are symmetric, then for any scalars \(k\) and \(m\), the combination \(kA + mB\) will also be symmetric because:
  \[
  (kA + mB)^T = kA^T + mB^T = kA + mB
  \]
  This holds since \(A^T = A\) and \(B^T = B\).
- **Answer: True**

### 5. **If \(A\) is a square matrix, then \(A + A^T\) is skew-symmetric.**
- A matrix \(M\) is skew-symmetric if \(M^T = -M\). However, for \(A + A^T\), we have:
  \[
  (A + A^T)^T = A^T + (A^T)^T = A + A^T
  \]
  Since \(A + A^T = (A + A^T)^T\), this shows that \(A + A^T\) is symmetric, not skew-symmetric.
- **Answer: False**

---

### Summary:
1. False
2. True
3. False
4. True
5. False

Would you like more details on any of these? Here are some related questions to expand on this topic:

1. What conditions must be met for a matrix to be symmetric?
2. What are the properties of a skew-symmetric matrix?
3. How does the transpose operation affect the determinant of a matrix?
4. What is the relationship between a matrix and its adjugate?
5. How do symmetric and skew-symmetric matrices relate to eigenvalues?

**Tip:** When dealing with matrix transposition, remember that transposing a product of matrices reverses their order: \((AB)^T = B^T A^T\).

Mark the following True or False:
1. If the (3, 1)-entry of a matrix A is 5, then the (1, 3)-entry of A^T is −5.
2. A and A^T have the same main diagonal for every matrix A.
3. If B is symmetric and A^T = 3B, then A = 3B.
4. If A and B are symmetric, then kA + mB is symmetric for any scalars k and m.
5. If A is a square matrix, then A + A^T is skew-symmetric.

Explore True/False questions related to matrix transposition and symmetry. Topics include properties of symmetric and skew-symmetric matrices, matrix addition, and the transpose operation. Perfect for undergraduate students studying linear algebra.

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