If a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements ?

Answer

To determine the possible orders of a matrix based on the number of elements it contains, we can use the formula for the total number of elements in a matrix, which is given by the product of its rows (m) and columns (n). Therefore, for a matrix with a total of \( E \) elements, we have:

\[ m \times n = E \]

### Step-by-step Solution:

1. **For a matrix with 18 elements:**
- We need to find pairs of integers \( (m, n) \) such that \( m \times n = 18 \).
- We can start by listing the factors of 18:
- \( 1 \times 18 = 18 \)
- \( 2 \times 9 = 18 \)
- \( 3 \times 6 = 18 \)
- \( 6 \times 3 = 18 \)
- \( 9 \times 2 = 18 \)
- \( 18 \times 1 = 18 \)
- Therefore, the possible orders of the matrix with 18 elements are:
- \( 1 \times 18 \)
- \( 2 \times 9 \)
- \( 3 \times 6 \)
- \( 6 \times 3 \)
- \( 9 \times 2 \)
- \( 18 \times 1 \)

2. **For a matrix with 5 elements:**
- Similarly, we find pairs of integers \( (m, n) \) such that \( m \times n = 5 \).
- The factors of 5 are:
- \( 1 \times 5 = 5 \)
- \( 5 \times 1 = 5 \)
- Therefore, the possible orders of the matrix with 5 elements are:
- \( 1 \times 5 \)
- \( 5 \times 1 \)

### Summary of Possible Orders:
- For 18 elements: \( 1 \times 18, 2 \times 9, 3 \times 6, 6 \times 3, 9 \times 2, 18 \times 1 \) (Total: 6 possible orders)
- For 5 elements: \( 1 \times 5, 5 \times 1 \) (Total: 2 possible orders)