If A is a square matrix, then

Options

A. AA' is symmetric
B. AA' is skew – symmetric
C. A'A is symmetric
D. A'A is skew – symmetric

Answer

To solve the problem, we need to analyze the properties of the products of a square matrix \( A \) and its transpose \( A' \). We will check whether the matrices \( AA' \) and \( A'A \) are symmetric or skew-symmetric.

### Step-by-Step Solution:

1. **Understanding the Matrix and Its Transpose**:
- Let \( A \) be a square matrix of order \( n \times n \).
- The transpose of \( A \), denoted as \( A' \) (or \( A^T \)), is obtained by interchanging the rows and columns of \( A \).

2. **Checking if \( AA' \) is Symmetric**:
- A matrix \( P \) is symmetric if \( P = P' \).
- We need to find \( (AA')' \):
\[ (AA')' = A'A' \]
- By the property of transposes, we know that \( (AB)' = B'A' \).
- Thus, we have:
\[ (AA')' = A'A \]
- Now we check if \( AA' = (AA')' \):
\[ AA' = A'A \]
- Since \( AA' \) is equal to its transpose, \( AA' \) is symmetric.

3. **Checking if \( A'A \) is Symmetric**:
- Now, we check \( (A'A)' \):
\[ (A'A)' = A'A' \]
- Again using the property of transposes:
\[ (A'A)' = A'A \]
- We check if \( A'A = (A'A)' \):
\[ A'A = A'A \]
- Since \( A'A \) is equal to its transpose, \( A'A \) is also symmetric.

4. **Conclusion**:
- From the above analysis, we conclude that:
- \( AA' \) is symmetric.
- \( A'A \) is symmetric.
- Therefore, the correct options are:
- \( AA' \) is symmetric (Option A).
- \( A'A \) is symmetric (Option C).

### Final Answer:
- **Correct Options**: A and C are correct.