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 possible orders of matrices A and B such that the product AB is defined.

### Step 1: Understanding Matrix Orders
A matrix A with \( m \) rows and \( n \) columns is said to have an order of \( m \times n \). The number of elements in matrix A is given by \( m \times n \).

### Step 2: Finding Possible Orders for Matrix A
Given that matrix A has 24 elements, we can find the possible pairs \( (m, n) \) such that \( m \times n = 24 \).

The factor pairs of 24 are:
- \( 1 \times 24 \)
- \( 2 \times 12 \)
- \( 3 \times 8 \)
- \( 4 \times 6 \)
- \( 6 \times 4 \)
- \( 8 \times 3 \)
- \( 12 \times 2 \)
- \( 24 \times 1 \)

Thus, the possible orders for matrix A are:
1. \( 1 \times 24 \)
2. \( 2 \times 12 \)
3. \( 3 \times 8 \)
4. \( 4 \times 6 \)
5. \( 6 \times 4 \)
6. \( 8 \times 3 \)
7. \( 12 \times 2 \)
8. \( 24 \times 1 \)

### Step 3: Finding Possible Orders for Matrix B
Similarly, for matrix B, which has 40 elements, we find the pairs \( (p, q) \) such that \( p \times q = 40 \).

The factor pairs of 40 are:
- \( 1 \times 40 \)
- \( 2 \times 20 \)
- \( 4 \times 10 \)
- \( 5 \times 8 \)
- \( 8 \times 5 \)
- \( 10 \times 4 \)
- \( 20 \times 2 \)
- \( 40 \times 1 \)

Thus, the possible orders for matrix B are:
1. \( 1 \times 40 \)
2. \( 2 \times 20 \)
3. \( 4 \times 10 \)
4. \( 5 \times 8 \)
5. \( 8 \times 5 \)
6. \( 10 \times 4 \)
7. \( 20 \times 2 \)
8. \( 40 \times 1 \)

### Step 4: Conditions for 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. Therefore, if matrix A has an order of \( m \times n \) and matrix B has an order of \( p \times q \), we need \( n = p \).

### Step 5: Counting Compatible Orders
Now, we will check each possible order of A against the orders of B to see which combinations satisfy the condition \( n = p \):

- For \( A: 1 \times 24 \) (n=24), no compatible B.
- For \( A: 2 \times 12 \) (n=12), compatible with \( B: 12 \times 2 \).
- For \( A: 3 \times 8 \) (n=8), compatible with \( B: 8 \times 5 \) and \( B: 5 \times 8 \).
- For \( A: 4 \times 6 \) (n=6), no compatible B.
- For \( A: 6 \times 4 \) (n=4), compatible with \( B: 4 \times 10 \) and \( B: 10 \times 4 \).
- For \( A: 8 \times 3 \) (n=3), no compatible B.
- For \( A: 12 \times 2 \) (n=2), compatible with \( B: 2 \times 20 \).
- For \( A: 24 \times 1 \) (n=1), compatible with \( B: 1 \times 40 \).

### Step 6: Total Compatible Orders
Now we count the compatible pairs:
1. \( (2 \times 12, 12 \times 2) \)
2. \( (3 \times 8, 8 \times 5) \)
3. \( (3 \times 8, 5 \times 8) \)
4. \( (6 \times 4, 4 \times 10) \)
5. \( (6 \times 4, 10 \times 4) \)
6. \( (12 \times 2, 2 \times 20) \)
7. \( (24 \times 1, 1 \times 40) \)

This gives us a total of 7 compatible orders for the matrices A and B such that the product AB is defined.

### Final Answer
The number of possible orders of A and B such that AB is defined is **7**.