Exercise 3.1 · Q3: Possible orders of a matrix with 18 or 5 elements

Basics you need

Part 1: 18 elements

1. Turn the question into an equation

   If the matrix is $m\times n$⁠, it has $mn$ elements, so

   $$mn = 18, \qquad m, n \in \N.$$

   Each solution $(m, n)$ is one possible order.

   This is the method of Example 2. Rows and columns are counts, so $m$ and $n$ must be natural numbers, and $n = 18/m$ must come out whole.
   An $m\times n$ matrix has $mn$ elements (Section 3.2.1). Basics: rows and columns

2. Try every number of rows

   Rows $m$	$18 \div m$	Whole number?	Order
   1	18	yes	$1\times18$
   2	9	yes	$2\times9$
   3	6	yes	$3\times6$
   4	4.5	no	none
   5	3.6	no	none
   6	3	yes	$6\times3$
   7, 8	between 2 and 3	no	none
   9	2	yes	$9\times2$
   10 to 17	between 1 and 2	no	none
   18	1	yes	$18\times1$

   No $m$ larger than 18 can work, since $n = 18/m$ would be less than 1.

   Checking every candidate makes the list complete. For $m = 7, 8$ we have $18/9 = 2 < 18/m < 18/6 = 3$⁠; for $10\le m\le17$ we have $1 < 18/m < 2$⁠. A number strictly between two consecutive whole numbers is not whole.
   $m$ divides 18 exactly when $18\div m$ is a whole number.

3. List the orders

   The divisors of 18 are $1, 2, 3, 6, 9, 18$⁠, so the possible orders are

   $$\begin{gathered} 1\times18,\ 2\times9,\ 3\times6, \\ 6\times3,\ 9\times2,\ 18\times1. \end{gathered}$$

   That is 6 possible orders.

   They come in mirror pairs ($1\times18$ and $18\times1$⁠, and so on), and both members of each pair are needed because they have different numbers of rows. 18 is not a perfect square, so no square order $k\times k$ appears.
   An order is an ordered pair (rows, columns). Lesson: order of a matrix

Part 2: 5 elements

1. Same equation, now with 5

   We need $mn = 5$ with $m, n\in\N$⁠. Testing $m = 2, 3, 4$ gives $2.5$⁠, $1.66\ldots$⁠, $1.25$⁠: none is whole. Only $m = 1$ and $m = 5$ work:

   $$\begin{aligned} m = 1 &\;\Rightarrow\; n = 5, \\ m = 5 &\;\Rightarrow\; n = 1. \end{aligned}$$

   So a matrix with 5 elements has order $1\times5$ or $5\times1$.

   5 is prime, so its only factor pairs are $1\times5$ and $5\times1$⁠. Such a matrix must be a single row (a row matrix) or a single column (a column matrix).
   Row matrices $1\times n$ and column matrices $m\times1$⁠: Section 3.3, types (i) and (ii). Lesson: types of matrices

Every fact used, and where it comes from