How many possible orders a matrix having 8 elements can have?

Options

A. 5
B. 2
C. 4
D. 3

Answer

To determine how many possible orders a matrix with 8 elements can have, we need to find all the combinations of rows and columns that multiply to give us 8.

### Step-by-Step Solution:

1. **Understanding Matrix Orders**: A matrix order is defined by the number of rows and columns it has. If a matrix has \( r \) rows and \( c \) columns, the total number of elements in the matrix is given by the product \( r \times c \).

2. **Setting Up the Equation**: For a matrix with 8 elements, we need to find pairs of \( (r, c) \) such that:
\[ r \times c = 8 \]

3. **Finding the Factor Pairs of 8**: We will list all pairs of positive integers \( (r, c) \) that satisfy the equation:
- \( 1 \times 8 \)
- \( 2 \times 4 \)
- \( 4 \times 2 \)
- \( 8 \times 1 \)

4. **Counting the Unique Orders**: From the pairs listed, we can see that:
- \( (1, 8) \) corresponds to a \( 1 \times 8 \) matrix.
- \( (2, 4) \) corresponds to a \( 2 \times 4 \) matrix.
- \( (4, 2) \) corresponds to a \( 4 \times 2 \) matrix.
- \( (8, 1) \) corresponds to an \( 8 \times 1 \) matrix.

5. **Total Unique Orders**: The unique orders we found are:
- \( 1 \times 8 \)
- \( 2 \times 4 \)
- \( 4 \times 2 \)
- \( 8 \times 1 \)

Therefore, there are a total of **4 possible orders** for a matrix with 8 elements.

### Final Answer:
The number of possible orders a matrix having 8 elements can have is **4**.