If A and B are matrices with 24 and 40 elements respectively, then the number of possible orders of A and B such that AB is defined is

Options

A. 2
B. 3
C. 4
D. 8

Answer

To solve the problem, we need to determine the number of possible orders of matrices A and B such that the product AB is defined.

### Step-by-Step Solution:

1. **Identify the number of elements in matrices A and B**:
- Matrix A has 24 elements.
- Matrix B has 40 elements.

2. **Express the orders of matrices A and B**:
- Let the order of matrix A be \( x \times n \), where \( x \) is the number of rows and \( n \) is the number of columns.
- Let the order of matrix B be \( n \times m \), where \( n \) is the number of rows and \( m \) is the number of columns.

3. **Set up the equations based on the number of elements**:
- For matrix A: \( x \cdot n = 24 \)
- For matrix B: \( n \cdot m = 40 \)

4. **Find the factors of 24**:
- The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

5. **Find the factors of 40**:
- The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40.

6. **Identify the common factors**:
- The common factors of 24 and 40 are: 1, 2, 4, 8.

7. **Count the number of common factors**:
- There are 4 common factors (1, 2, 4, 8).

8. **Conclusion**:
- The number of possible orders of matrices A and B such that the product AB is defined is 4.

### Final Answer:
The number of possible orders of A and B such that AB is defined is **4**.

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