A and B are tow square matrices of same order and A' denotes the transpose of A, then

Options

A. `(AB)'=B'A'`
B. `(AB)' =A'B'`
C. `AB=0rArrabsA=0orabsB=O`
D. `AB =0rArrA=0orB=0`

Answer

To solve the problem involving two square matrices \( A \) and \( B \) of the same order, we need to analyze the given statements regarding their properties, particularly focusing on the transpose of their product. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Matrices**:
Let \( A \) and \( B \) be two square matrices of order \( n \). We denote the transpose of matrix \( A \) as \( A' \) (or \( A^T \)) and the transpose of matrix \( B \) as \( B' \) (or \( B^T \)).

2. **Statement 1: \( (AB)' = B'A' \)**:
- To verify this statement, we need to compute the transpose of the product \( AB \).
- The transpose of a product of two matrices is given by the formula:
\[ (AB)' = B'A' \]
- This property holds true for any two matrices \( A \) and \( B \) of the same order. Thus, this statement is **true**.

3. **Statement 2: \( (AB)' = A'B' \)**:
- We will check if the transpose of the product \( AB \) equals \( A'B' \).
- According to the transpose property:
\[ (AB)' = B'A' \]
- Since \( A'B' \) is not equal to \( B'A' \) in general, this statement is **false**.

4. **Statement 3: If \( AB = 0 \), then \( \text{det}(A) = 0 \) and \( \text{det}(B) = 0 \)**:
- The product of two matrices being the zero matrix does not imply that either matrix must be singular (determinant zero).
- For example, if \( A \) is a non-zero matrix and \( B \) is a zero matrix, then \( AB = 0 \) but \( \text{det}(A) \neq 0 \). Thus, this statement is **false**.

5. **Statement 4: If \( AB = 0 \), then \( A = 0 \) and \( B = 0 \)**:
- Similar to the previous statement, \( AB = 0 \) does not imply that both \( A \) and \( B \) must be zero matrices.
- Again, if \( A \) is a non-zero matrix and \( B \) is a zero matrix, then \( AB = 0 \) holds true. Therefore, this statement is also **false**.

### Conclusion:
The only correct statement is:
- \( (AB)' = B'A' \)

### Final Answer:
The correct option is **Statement 1**.