If a matrix has 12 elements, what are the possible orders it can have?

Options

A. 6
B. 4
C. 2
D. 8

Explanation

To determine the possible orders of a matrix that contains 12 elements, we need to express the order of the matrix in the form of \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. The product of \( m \) and \( n \) must equal the total number of elements, which is 12.

### Step-by-Step Solution:

1. **Understanding Matrix Order**:
The order of a matrix is represented as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. The total number of elements in the matrix is given by the product \( m \times n \).

2. **Setting Up the Equation**:
Since the matrix has 12 elements, we set up the equation:
\[ m \times n = 12 \]

3. **Finding Factor Pairs**:
To find the possible values of \( m \) and \( n \), we need to find all pairs of positive integers whose product is 12. We can do this by determining the factors of 12.

4. **Listing the Factors of 12**:
The factors of 12 are:
- 1
- 2
- 3
- 4
- 6
- 12

5. **Forming Factor Pairs**:
Now, we can form pairs \( (m, n) \) from these factors:
- \( (1, 12) \)
- \( (2, 6) \)
- \( (3, 4) \)
- \( (4, 3) \)
- \( (6, 2) \)
- \( (12, 1) \)

6. **Listing Possible Orders**:
The possible orders of the matrix are:
- \( 1 \times 12 \)
- \( 2 \times 6 \)
- \( 3 \times 4 \)
- \( 4 \times 3 \)
- \( 6 \times 2 \)
- \( 12 \times 1 \)

### Conclusion:
The possible orders of a matrix with 12 elements are:
- \( 1 \times 12 \)
- \( 2 \times 6 \)
- \( 3 \times 4 \)
- \( 4 \times 3 \)
- \( 6 \times 2 \)
- \( 12 \times 1 \)