Q.If a matrix has $8$ elements, what are the possible orders it can have?

The key idea is that the order of a matrix is given by $m \times n$, where $m$ is the number of rows and $n$ is the number of columns. The total number of elements is the product $m \times n$.

Reasoning:

1. We need all pairs of positive integers $(m, n)$ such that $m \times n = 8$. …

The key idea is simple: a matrix’s order is defined by its number of rows and columns. If a matrix has $8$ elements total, then the product of its rows and columns must be $8$. So we are really just asking: in how many ways can we write $8$ as a product of two positive integers (where order matters, because a $2 \times 4$ matrix is different from a $4 \times 2$ matrix)?

Let’s walk through it.

1. Understand what “order” means.

   The order of a matrix is written as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns. The total number of elements is $m \times n$. So if a matrix has $8$ elements, we need $m \times n = 8$, with $m$ and $n$ both being positive integers.

2. List all factor pairs of 8.

   The positive integer factor pairs of 8 are:

   • $1 \times 8$
   • $2 \times 4$
   • $4 \times 2$
   • $8 \times 1$

   Notice that $m$ and $n$ are not interchangeable in general — a $2 \times 4$ matrix has 2 rows and 4 columns, while a $4 \times 2$ matrix has 4 rows and 2 columns. Both are valid and distinct orders.

3. Check for any other possibilities.

   Could $m$ or $n$ be something like $\frac{1}{2}$? No — rows and columns must be whole numbers. Could we have $m = 8$, $n = 1$? Yes, that’s a row matrix (or row vector). Could we have $m = 1$, $n = 8$? Yes, that’s a column matrix. Both are perfectly valid matrices. …

Method: Listing possible orders from the number of elements

Use this whenever you are told a matrix has $N$ elements and must list every possible order.

Steps

Step 1: Use the fact that order $\times$ = number of elements.

An $m\times n$ matrix has exactly $m\cdot n$ entries, so you need every ordered pair of positive integers $(m,n)$ with $m\cdot n = N$.

Step 2: List all factor pairs of $N$ — order matters. …

Common Mistakes

Mistake 1: Forgetting the $1\times N$ and $N\times1$ orders.

Why it's wrong: a matrix can have a single row or column, so $1\times8$ and $8\times1$ are valid. Correct approach: include the extreme factor pairs, not just the "middle" ones.

Mistake 2: Treating $m\times n$ and $n\times m$ as the same order.

Why it's wrong: a $2\times4$ matrix differs from a $4\times2$ one; both are distinct valid orders. Correct approach: count ordered factor pairs. …