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Chapter 3 · Matrices

Example 2: Possible orders of a matrix with 8 elements

ExamplesOrder of a matrix · Section 3.2.1
The question

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

Exam answer

Let the order be m×nm\times n⁠. Then mn=8mn = 8⁠, where m,n∈Nm, n \in \N⁠.

8=1×8=8×1=2×4=4×2\begin{aligned} 8 &= 1\times8 = 8\times1 \\ &= 2\times4 = 4\times2 \end{aligned}

∴\therefore the possible orders are 1×81\times8⁠, 8×18\times1⁠, 2×42\times4⁠, 4×24\times2⁠.

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

The one idea this problem is built on

A matrix of order m×nm\times n has mm rows of nn entries each, so it has exactly mnmn elements. "8 elements" therefore means mn=8mn = 8⁠, where mm and nn are natural numbers (you cannot have 2.5 rows). Finding the orders is the same as finding every way to write 8 as a product of two natural numbers, in order.

Given, to find, and the plan
Given
A matrix with 8 elements.
To find
Every order m×nm\times n it could have.
Plan
Turn "8 elements" into the equation mn=8mn = 8⁠. Try every possible number of rows m=1,2,…,8m = 1, 2, \dots, 8⁠; for each, see whether n=8/mn = 8/m is a natural number. Collect the ordered pairs (m,n)(m, n)⁠.

Basics you need

1Order and number of elements

An m×nm\times n matrix has mm rows and nn columns. Each row has nn entries, and there are mm rows, so there are m×nm\times n entries in total. For example a 3×23\times2 matrix has 6 entries. Basics: rows, columns and the address a_ij

2Rows and columns are counted by natural numbers

mm and nn are each at least 1 and are whole numbers: a matrix cannot have 0 rows or half a column.

3Order matters

2×42\times4 (2 rows, 4 columns) and 4×24\times2 (4 rows, 2 columns) are different shapes, so they count as different orders, even though both have 8 entries.

Solution

  1. Turn "8 elements" into an equation

    Suppose the matrix has order m×nm\times n⁠. It then has mnmn elements, so

    mn=8,m,n∈N.mn = 8, \qquad m, n \in \N.
    The question gives a count of entries and asks for a shape. The link between the two is the fact that an m×nm\times n matrix has mnmn entries (mm rows of nn each), so we write that link as an equation.
    Number of elements of an m×nm\times n matrix =mn= mn (Section 3.2.1). Basics: rows and columns

  2. Try every possible number of rows

    Since n≥1n \ge 1⁠, we have m=8/n≤8m = 8/n \le 8⁠, so mm can only be one of 1,2,…,81, 2, \dots, 8⁠. For each, n=8/mn = 8/m must be a natural number:

    Rows mmn=8/mn = 8/mNatural number?Order
    18yes1×81\times8
    24yes2×42\times4
    38/38/3nonone
    42yes4×24\times2
    58/58/5nonone
    64/34/3nonone
    78/78/7nonone
    81yes8×18\times1
    Checking all 8 values of mm guarantees we miss nothing, and the bound m≤8m\le 8 guarantees there is nothing beyond the table. The values that work are exactly the divisors of 8, namely 1, 2, 4, 8, and each one gives the partner n=8/mn = 8/m⁠.
    n=8/mn = 8/m is a natural number exactly when mm divides 8. The divisors of 8 are 1, 2, 4, 8. Basics: solving equations

  3. List the orders as ordered pairs

    The ordered pairs (m,n)(m, n) with mn=8mn = 8 are

    (1,8),(8,1),(2,4),(4,2).(1, 8),\quad (8, 1),\quad (2, 4),\quad (4, 2).

    So the possible orders are 1×8, 8×1, 2×4, 4×21\times8,\ 8\times1,\ 2\times4,\ 4\times2.

    1 × 8 2 × 4 4 × 2 8 × 1 order = (number of rows) × (number of columns); 8 dots every time
    The same 8 entries can be laid out in exactly four rectangles. The 1×81\times8 one is a row matrix, the 8×18\times1 one is a column matrix.
    The pairs are ordered because the first number counts rows and the second counts columns. 2×42\times4 and 4×24\times2 are different matrices in shape, so both must be listed. There is no square order m×mm\times m⁠, because that would need m2=8m^2 = 8⁠, and 8 is not a perfect square.
    A 1×n1\times n matrix is a row matrix and an m×1m\times1 matrix is a column matrix (Section 3.3). Lesson: types of matrices
Result

A matrix with 8 elements can have order 1×81\times8⁠, 8×18\times1⁠, 2×42\times4 or 4×24\times2, and no other. This agrees with the textbook's solution.

Check it with numbers

1×8=81\times8 = 8⁠, 8×1=88\times1 = 8⁠, 2×4=82\times4 = 8⁠, 4×2=84\times2 = 8⁠: each order really has 8 entries. A near miss such as 3×33\times3 has 9 entries and 3×23\times2 has 6, so neither is possible.

Common mistakes
  1. Listing only 2×42\times4 and 1×81\times8⁠. The reversed orders 4×24\times2 and 8×18\times1 are different shapes and must be included.
  2. Forgetting the trivial factor 1. 1×81\times8 and 8×18\times1 (a single row, a single column) are perfectly good matrices.
  3. Adding instead of multiplying. The number of elements is m×nm\times n⁠, not m+nm+n⁠.
What to take away
  1. An m×nm\times n matrix has mnmn elements; to find the orders for a given count, list every ordered factor pair of that count.
  2. The number of possible orders equals the number of divisors of the count. For 8 that is 4; for a prime such as 13 it is only 2 (1×131\times13 and 13×113\times1⁠).

Every fact used, and where it comes from

FactUsed inSource
An m×nm\times n matrix has mnmn elementsStep 1Section 3.2.1 (Lesson: what a matrix is)
m,nm, n are natural numbersSteps 1, 2Section 3.2.1, 1≤i≤m1\le i\le m⁠, 1≤j≤n1\le j\le n⁠, i,j∈Ni, j\in\N
Divisors of 8 are 1, 2, 4, 8Step 2Arithmetic
Row matrix 1×n1\times n⁠, column matrix m×1m\times1Step 3Section 3.3 (i), (ii) (Lesson: types of matrices)
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