Possible Orders of a Matrix with Given Number of Elements

Concept Used

If a matrix has order \( m \times n \), then the number of elements = \( m \times n \). :contentReference[oaicite:0]{index=0}

So, to find possible orders, we factorize the total number of elements into all possible products.

Case 1: Matrix with 8 Elements

We find all factor pairs of 8:

\[ 8 = 1 \times 8 = 2 \times 4 = 4 \times 2 = 8 \times 1 \]

Possible Orders:

• \( 1 \times 8 \)
• \( 2 \times 4 \)
• \( 4 \times 2 \)
• \( 8 \times 1 \)

Case 2: Matrix with 5 Elements

Since 5 is a prime number, its only factors are:

\[ 5 = 1 \times 5 = 5 \times 1 \]

Possible Orders:

• \( 1 \times 5 \)
• \( 5 \times 1 \)

Final Answer

For 8 elements: \( (1 \times 8), (2 \times 4), (4 \times 2), (8 \times 1) \)

For 5 elements: \( (1 \times 5), (5 \times 1) \)