Chapter 3 · Matrices

Exercise 3.1 · Q2: Possible orders of a matrix with 24 or 13 elements

Exercise 3.1Order of a matrix · Section 3.2

The question

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Exam answer

A matrix of order $m\times n$ has $mn$ elements.

24 elements: $mn = 24$ ⁠, and

\begin{aligned} 24 &= 1\times24 = 2\times12 \\ &= 3\times8 = 4\times6 \end{aligned}

Possible orders: $1\times24$ ⁠, $2\times12$ ⁠, $3\times8$ ⁠, $4\times6$ ⁠, $6\times4$ ⁠, $8\times3$ ⁠, $12\times2$ ⁠, $24\times1$ ⁠.

13 elements: $mn = 13$ ⁠. As 13 is prime, $13 = 1\times13$ only.

Possible orders: $1\times13$ ⁠, $13\times1$ ⁠.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

A matrix of order $m\times n$ has exactly $mn$ elements. So "a matrix with 24 elements" means $mn = 24$ ⁠, where $m$ (rows) and $n$ (columns) are natural numbers. Finding the possible orders is the same as finding every ordered pair of natural numbers whose product is 24. Ordered, because $3\times8$ (3 rows) and $8\times3$ (8 rows) are different shapes.

Given, to find, and the plan

Given: A matrix with 24 elements; separately, a matrix with 13 elements.
To find: All possible orders $m\times n$ in each case.
Plan: (1) Turn "24 elements" into the equation $mn = 24$ with $m, n \in \N$ ⁠. (2) Try every possible number of rows $m$ from 1 to 24 systematically, keeping those that divide 24. (3) Repeat for 13.

Basics you need

1Order and number of elements

An $m\times n$ matrix has $m$ rows and $n$ columns, hence $mn$ elements. Both $m$ and $n$ are natural numbers: a matrix cannot have 0 rows or $2.5$ rows. Basics: rows, columns and the address

2Divisors

$d$ divides $N$ when $N = d\cdot k$ for a natural number $k$ ⁠. For example 4 divides 24 since $24 = 4\cdot6$ ⁠, but 5 does not, since $24 \div 5 = 4.8$ ⁠.

3Prime numbers

A prime has exactly two divisors, 1 and itself. 13 is prime: none of 2, 3, …, 12 divides it.

4Order is an ordered pair

$m\times n$ and $n\times m$ are different orders when $m\ne n$ ⁠: one is tall, the other wide. Lesson: what a matrix is

Part 1: 24 elements

Turn the question into an equation

Suppose the matrix has order $m\times n$ ⁠. It has $mn$ elements, so
$mn = 24, \qquad m, n \in \N.$
Each solution $(m, n)$ gives one possible order, and each possible order gives one solution.

This is exactly the method of Example 2 (8 elements). Writing the condition as an equation turns "what shapes are possible?" into "which pairs of natural numbers multiply to 24?", which we can answer completely.

An $m\times n$ matrix has $mn$ elements (Section 3.2.1). Basics: rows and columns

Try every number of rows

The number of rows $m$ must divide 24 exactly, because $n = 24/m$ has to be a natural number. Test every $m$ from 1 to 24:

Rows $m$	$24 \div m$	Whole number?	Order
1	24	yes	$1\times24$
2	12	yes	$2\times12$
3	8	yes	$3\times8$
4	6	yes	$4\times6$
5	4.8	no	none
6	4	yes	$6\times4$
7	3.43 (approx.)	no	none
8	3	yes	$8\times3$
9, 10, 11	between 2 and 3	no	none
12	2	yes	$12\times2$
13 to 23	between 1 and 2	no	none
24	1	yes	$24\times1$

No $m$ larger than 24 can work, since then $n = 24/m$ would be less than 1.

Testing every candidate is what makes the list complete; guessing a few factor pairs could miss one. For

9\le m\le 11

we have

24/12 = 2 < 24/m < 24/8 = 3

⁠, and for

13\le m\le23

we have

1 < 24/m < 2

⁠, so in both ranges

24/m

lies strictly between two consecutive whole numbers and cannot be a natural number.

m

divides 24 exactly when

24 \div m

is a whole number.

List the orders

The divisors of 24 are $1, 2, 3, 4, 6, 8, 12, 24$ ⁠, each giving one order:
$\begin{gathered} 1\times24,\ 2\times12,\ 3\times8,\ 4\times6, \\ 6\times4,\ 8\times3,\ 12\times2,\ 24\times1. \end{gathered}$
So a matrix with 24 elements can have 8 possible orders, those listed above.

Three of the eight shapes with 24 entries. $3\times8$ and $8\times3$ use the same factor pair but are different orders: one has 3 rows, the other 8.

The pairs come in mirror images: $1\times24$ and $24\times1$ ⁠, $2\times12$ and $12\times2$ ⁠, and so on. Both members of each pair must be listed, because a $2\times12$ matrix (2 rows) and a $12\times2$ matrix (12 rows) have different orders. Since 24 is not a perfect square, no order of the form $k\times k$ appears, so there is no square matrix with 24 elements.

An order is an ordered pair (rows, columns). Lesson: order of a matrix

Part 2: 13 elements

Same equation, now with 13

Now we need $mn = 13$ with $m, n \in \N$ ⁠. The number of rows $m$ must divide 13. Testing $m = 2, 3, \dots, 12$ ⁠: none of them divides 13 (the quotients $6.5, 4.33\ldots, 3.25, 2.6, \dots, 1.08\ldots$ are not whole), because 13 is prime. Only $m = 1$ and $m = 13$ work:
$\begin{aligned} m = 1 &\;\Rightarrow\; n = 13, \\ m = 13 &\;\Rightarrow\; n = 1. \end{aligned}$
So a matrix with 13 elements can only be of order $1\times13$ or $13\times1$ .

A prime number has no factor pairs other than $1\times p$ and $p\times1$ ⁠. In matrix language: a matrix with a prime number of elements must be a single row (a row matrix, $1\times13$ ⁠) or a single column (a column matrix, $13\times1$ ⁠).

13 is prime: its only divisors are 1 and 13. Row and column matrices are types (i) and (ii) in Section 3.3. Lesson: types of matrices

Result

With 24 elements: $1\times24$ ⁠, $2\times12$ ⁠, $3\times8$ ⁠, $4\times6$ ⁠, $6\times4$ ⁠, $8\times3$ ⁠, $12\times2$ ⁠, $24\times1$ (8 orders). With 13 elements: $1\times13$ ⁠, $13\times1$ (2 orders). (Textbook answer: the same.)

Check it with numbers

Multiply each pair back: $1\cdot24 = 2\cdot12 = 3\cdot8 = 4\cdot6 = 24$ ✓, and the reversed pairs give the same products ✓. The number of divisors of 24 can also be counted from $24 = 2^3\cdot3$ ⁠: there are $(3+1)(1+1) = 8$ divisors, matching the 8 orders ✓. For $13 = 13^1$ ⁠: $1+1 = 2$ divisors, matching 2 orders ✓.

Common mistakes

Listing only $1\times24, 2\times12, 3\times8, 4\times6$ ⁠. The reversed orders are different matrices and must be included.
Missing a divisor, for example $12\times2$ ⁠. Test every possible number of rows systematically.
Including non-whole orders such as $5\times4.8$ ⁠. Rows and columns are counted, so both must be natural numbers.
Saying "13 elements is impossible". A single row or a single column of 13 entries is a perfectly good matrix.

What to take away

Possible orders of a matrix with $N$ elements = ordered pairs $(m, n)$ of natural numbers with $mn = N$ ⁠.
The number of possible orders equals the number of divisors of $N$ ⁠.
If $N$ is prime, the only orders are $1\times N$ and $N\times1$ ⁠.

Every fact used, and where it comes from

Fact	Used in	Source
An $m\times n$ matrix has $mn$ elements	Part 1, Step 1; Part 2, Step 1	Section 3.2.1 (Lesson: what a matrix is)
Possible orders are the ordered pairs with product equal to the number of elements	Part 1, Steps 1–3	Example 2
Row matrix $1\times n$ ⁠, column matrix $m\times1$	Part 2, Step 1	Section 3.3, types (i) and (ii)
13 is prime; $24 = 2^3\cdot3$ has 8 divisors	Part 2, Step 1; Check	Arithmetic, recomputed in the site's verification script

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Exercise 3.1 · Q2: Possible orders of a matrix with 24 or 13 elements

Basics you need

Part 1: 24 elements

Turn the question into an equation

Try every number of rows

List the orders

Part 2: 13 elements

Same equation, now with 13

Every fact used, and where it comes from