If A and B are two matrices such that A+B and AB are both defined, then

Options

A. A and B are two matrices not necessarily of same order
B. A and B are square matrices of same order
C. Number of colomns of A=number of rows of B.
D. None of these

Answer

To solve the problem, we need to analyze the conditions under which the matrices A and B can be added (A + B) and multiplied (AB).

### Step-by-Step Solution:

1. **Understanding Matrix Addition**:
- For two matrices A and B to be added (A + B), they must have the same dimensions. This means that if A is of order \( m \times n \), then B must also be of order \( m \times n \).

2. **Understanding Matrix Multiplication**:
- For the product of two matrices A and B (AB) to be defined, the number of columns in A must equal the number of rows in B. If A is of order \( m \times n \) and B is of order \( p \times q \), then for the multiplication AB to be defined, we must have \( n = p \).

3. **Combining the Conditions**:
- Since both A + B and AB are defined, we can combine the conditions:
- From the addition, we have that \( A \) and \( B \) must have the same order, say \( m \times n \).
- From the multiplication, we have that the number of columns of A (which is \( n \)) must equal the number of rows of B (which is \( m \)).

4. **Conclusion**:
- Therefore, if A and B are both \( m \times n \) matrices, then for the multiplication to be defined, we must have \( n = m \). This leads us to conclude that both matrices must be square matrices of the same order.

### Final Statement:
If A and B are two matrices such that A + B and AB are both defined, then the number of columns of A is equal to the number of rows of B, and both matrices must be square matrices of the same order.