Which one of the following is wrong?

Options

A. (a0 The elements on the main diagonal of a symmetric matrix are all zero
B. (b) The elements on the main diagonal of a skew-symmetric amtrix are all zero
C. (c) For any square matrix `A,A A'` is symmetric
D. (d) For any square matrix `A,(A+A')^(2)=A^(2)+(A')^(2)+2A A'`

Answer

To solve the problem of identifying which statement is wrong among the given options regarding matrices, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1:** The elements on the main diagonal of a symmetric matrix are all zero.

**Analysis:**
- A symmetric matrix \( A \) satisfies the condition \( A^T = A \).
- The elements on the main diagonal are denoted as \( a_{ii} \).
- For a symmetric matrix, we have \( a_{ij} = a_{ji} \).
- For diagonal elements, \( i = j \), hence \( a_{ii} = a_{ii} \) which does not imply that \( a_{ii} \) must be zero.

**Conclusion:** This statement is **wrong**.

### Step 2: Analyze Statement 2
**Statement 2:** The elements on the main diagonal of a skew-symmetric matrix are all zero.

**Analysis:**
- A skew-symmetric matrix \( A \) satisfies the condition \( A^T = -A \).
- For diagonal elements, we have \( a_{ii} = -a_{ii} \).
- This implies \( 2a_{ii} = 0 \) or \( a_{ii} = 0 \).

**Conclusion:** This statement is **correct**.

### Step 3: Analyze Statement 3
**Statement 3:** For any square matrix \( A \), \( AA^T \) is symmetric.

**Analysis:**
- We need to check if \( (AA^T)^T = AA^T \).
- Using the property of transposes, \( (AB)^T = B^T A^T \), we have:
\[ (AA^T)^T = (A^T)^T A^T = A A^T \]
- Thus, \( AA^T \) is indeed symmetric.

**Conclusion:** This statement is **correct**.

### Step 4: Analyze Statement 4
**Statement 4:** For any square matrix \( A \), \( A + A^T \) is equal to \( (A + A^T)^2 \).

**Analysis:**
- We need to check if \( A + A^T = (A + A^T)^2 \).
- Expanding \( (A + A^T)^2 \):
\[ (A + A^T)(A + A^T) = A^2 + AA^T + A^TA + (A^T)^2 \]
- This expression does not simplify to \( A + A^T \) unless \( A \) is symmetric, which is not true for all matrices.

**Conclusion:** This statement is **wrong**.

### Final Conclusion
The wrong statements are **Statement 1** and **Statement 4**.