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

Options

A. A & B are two matrices not necessarily of same order
B. A and B are square matrices of same order
C. number of columns 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 and multiplied.

### Step-by-Step Solution:

1. **Understanding Matrix Addition**:
For the sum \( A + B \) to be defined, both matrices must have the same dimensions. This means that the number of rows and columns in matrix A must equal the number of rows and columns in matrix B.

**Hint**: Check the dimensions of both matrices for addition.

2. **Understanding Matrix Multiplication**:
For the product \( AB \) to be defined, the number of columns in matrix A must equal the number of rows in matrix B. This means if A is of size \( m \times n \), then B must be of size \( n \times p \) for some integer p.

**Hint**: Look at the relationship between the columns of A and the rows of B for multiplication.

3. **Analyzing the Options**:
- **Option A**: "A and B are two matrices not necessarily of the same order."
- This option is incorrect because if A and B are not of the same order, \( A + B \) cannot be defined.

- **Option B**: "A and B are square matrices of the same order."
- This option is correct because square matrices of the same order satisfy both conditions for addition and multiplication.

- **Option C**: "The number of columns of A is equal to the number of rows of B."
- This option is also incorrect because it only satisfies the condition for multiplication, not for addition.

- **Option D**: "None of these."
- This option is incorrect because we have already identified Option B as a valid condition.

4. **Conclusion**:
The correct answer is **Option B**, which states that A and B are square matrices of the same order.

### Final Answer:
**Option B**: A and B are square matrices of the same order.