Consider the following in respect of matrices A, B and C of same order :
1. `(A+B+C)'=A'+B'+C'`
2. `(AB)'=AB'`
3. `(ABC)'=C'B'A'`
Where A' is the transpose of the matrix A. Which of the above are correct ?

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Explanation

To determine which of the statements regarding matrices A, B, and C are correct, we will analyze each statement based on the properties of matrix transposition.

### Step-by-Step Solution:

1. **Statement 1: \((A + B + C)' = A' + B' + C'\)**

- **Explanation**: This statement is true based on the property of transposes. The transpose of a sum of matrices is equal to the sum of their transposes.
- **Verification**:
\[ (A + B + C)' = A' + B' + C' \]
- **Conclusion**: This statement is **correct**.

2. **Statement 2: \((AB)' = AB'\)**

- **Explanation**: This statement is false. The property of transposes states that the transpose of a product of two matrices is equal to the product of their transposes in reverse order.
- **Verification**:
\[ (AB)' = B'A' \]
- **Conclusion**: This statement is **incorrect**.

3. **Statement 3: \((ABC)' = C'B'A'\)**

- **Explanation**: This statement is true. Similar to the previous property, the transpose of a product of matrices is equal to the product of their transposes in reverse order.
- **Verification**:
\[ (ABC)' = C'B'A' \]
- **Conclusion**: This statement is **correct**.

### Final Conclusion:
- Statements 1 and 3 are correct.
- Statement 2 is incorrect.

Thus, the correct statements are **1 and 3**.