Q.If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

Concept: Matrix Order Possibilities — 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 $m \times n$. So possible orders are all factor pairs of the given number.

For 28 elements:

Factor pairs of 28: $(1, 28), (2, 14), (4, 7), (7, 4), (14, 2), (28, 1)$.

Thus possible orders: $1 \times 28$, $2 \times 14$, $4 \times 7$, $7 \times 4$, $14 \times 2$, $28 \times 1$.

For 13 elements:

13 is prime; its only factor pairs are $(1, 13)$ and $(13, 1)$.

Thus possible orders: $1 \times 13$ and $13 \times 1$.

A matrix is defined by its number of rows and columns. The total number of elements in a matrix is simply the product of its number of rows and number of columns. So if a matrix has $m$ rows and $n$ columns, it has $m \times n$ elements. The question asks: given a fixed total number of elements, what pairs $(m, n)$ of positive integers multiply to that total? Each such pair is a possible order.

The key insight is that we are looking for all factor pairs of the given number. Both $m$ and $n$ must be positive integers (a matrix cannot have zero rows or columns). The order is written as $m \times n$, and note that $m \times n$ and $n \times m$ are considered different orders unless $m = n$, because a matrix with 2 rows and 3 columns is not the same shape as one with 3 rows and 2 columns.

Let's work through each case.

1. For 28 elements: We need all positive integer pairs $(m, n)$ such that $m \times n = 28$. First, list all factor pairs of 28. The factors of 28 are 1, 2, 4, 7, 14, and 28. Pair them:

   • $1 \times 28 = 28$
   • $2 \times 14 = 28$
   • $4 \times 7 = 28$
   • $7 \times 4 = 28$
   • $14 \times 2 = 28$
   • $28 \times 1 = 28$

   So there are six possible orders: $1 \times 28$, $2 \times 14$, $4 \times 7$, $7 \times 4$, $14 \times 2$, and $28 \times 1$.

2. For 13 elements: 13 is a prime number. Its only positive factors are 1 and 13. So the only factor pairs are:

   • $1 \times 13 = 13$
   • $13 \times 1 = 13$

   Hence, only two possible orders: $1 \times 13$ and $13 \times 1$.

Method: Possible orders from a given number of elements

Use this whenever you are told how many elements a matrix has and asked for its possible orders.

Steps

Step 1: Recall the link between order and element count.

A matrix of order $m\times n$ has exactly $m\times n$ elements, with $m,n$ positive integers.

Step 2: List every ordered factor pair.

Find all pairs $(m,n)$ of positive integers whose product is the given total $N$. Because a $2\times14$ matrix differs from a $14\times2$ one, count each factor and its partner separately (unless $m=n$).

Step 3: Watch for primes.

If $N$ is prime, its only factor pairs are $1\times N$ and $N\times1$, so exactly two orders are possible.

Common Mistakes

Mistake 1: Listing only "half" the factor pairs.

Why it's wrong: $4\times7$ and $7\times4$ are different orders, so both count; missing the reversed pairs gives only 3 orders for 28 instead of 6. Correct approach: for each factor pair, include both arrangements (unless the two numbers are equal).

Mistake 2: Thinking a prime has many possible orders.

Why it's wrong: 13 is prime, so its only factorisations are $1\times13$ and $13\times1$ — just two orders. Correct approach: factor the number fully first.

Mistake 3: Allowing zero or non-integer dimensions.

Why it's wrong: rows and columns must be positive integers. Correct approach: only use whole-number factor pairs.