Year 11 General Matrices And Matrix Arithmetic

Using Matrices To Model (Represent) Practical Situations

5 practice questions 2 video lessons Theory + worked examples

Theory

A matrix is a tidy way to organise data that falls into a grid — like sales by store and product, or scores by student and test. Build a matrix by choosing what rows and columns represent. Read values using $M_{i j}$ notation. Row sums and column sums often carry real-world meaning. Special types include network and transition matrices.

A matrix is a tidy way to organise data that falls naturally into a grid. The three key skills are building a matrix from a context, reading information out of one, and interpreting entries and totals.

A row sum is the total across all entries in a single row — usually the total for one "thing" across all categories. A column sum is the total down a single column — usually the total for one category across all things.

A network (connection) matrix records the number of direct connections between points. For $n$ towns joined by roads, an $n \times n$ matrix has entry $(i, j)$ equal to the number of direct roads from town $i$ to town $j$ . Two-way networks give symmetric matrices; diagonal entries are usually 0.

A transition matrix describes movement between categories. Entry $(i, j)$ is the proportion that moves from category $j$ to category $i$ . Each column sums to 1.

The first diagram shows a sales matrix with row and column sums interpreted in their real-world contexts. The second shows a network of three towns and its corresponding connection matrix.

Row sum = Jan total across products. Column sum = Product B total across months.

A network of three towns and its symmetric

3 \times 3

connection matrix.

There are no calculation formulas here — just templates for the common contexts in which matrices appear.

Context	Rows represent	Columns represent	Entry $M_{i j}$ is
Pricing	drink type	size	price
Sales	store or month	product	units sold
Test scores	student	test	score
Inventory	store	product	quantity in stock
Network	start point	end point	number of direct connections
Transition	destination category	source category	proportion moving $j \to i$

Sums to know. Row sum — total for one row's "thing" across all categories. Column sum — total for one category across all "things". Always describe which sum and what it represents.

How to build a matrix from a context

Decide what the rows represent — usually the "things" being measured (stores, students, towns).
Decide what the columns represent — usually the categories (products, tests, destinations).
Fill in each entry as the value at that row/column intersection.
State the order as $m \times n$ = $m$ rows by $n$ columns.

How to read a value from a matrix

Identify which row corresponds to the "thing" you want.
Identify which column corresponds to the category you want.
Read the entry at that intersection. It is $M_{i j}$ where $i$ is the row number and $j$ is the column number.

How to build a network matrix

Label the rows and columns with the same set of names (towns, websites, people).
For each pair $(i, j)$ , count the number of direct connections from $i$ to $j$ and put that number in the entry.
Set the diagonal entries to 0 unless self-loops are allowed.
For two-way connections, the matrix is symmetric — entry $(i, j)$ equals entry $(j, i)$ .

EXAMPLE 1 — BUILD FROM DATA

A small business sells

120

of Product A,

85

of B, and

64

of C in January, and

140

of A,

92

of B,

78

of C in February. Build a matrix

S

with rows representing months and columns representing products.

SOLUTION

Row 1 holds January's figures, row 2 holds February's. Columns are A, B, C.

$S = [\begin{matrix} 120 & 85 & 64 \\ 140 & 92 & 78 \end{matrix}]$

The matrix has $2$ rows and $3$ columns, so its order is $2 \times 3$ .

Answer: $S$ is the $2 \times 3$ matrix above.

S = [\begin{matrix} 120 & 85 & 64 \\ 140 & 92 & 78 \end{matrix}]

EXAMPLE 2 — READ A VALUE

For the sales matrix

S

in Example 1, how many units of Product B were sold in February?

SOLUTION

February is row 2, Product B is column 2, so look up the entry $S_{22}$ .

S_{22}

=

92

Answer: $92$ units of Product B were sold in February.

S_{22} = 92

EXAMPLE 3 — ROW SUM AND ITS MEANING

A test-score matrix records four students' marks across four tests. Alice's row is

[\begin{matrix} 78 & 85 & 72 & 90 \end{matrix}]

. Compute her row sum and explain what it represents.

SOLUTION

Add the four entries in Alice's row.

Row sum	$=$	$78 + 85 + 72 + 90$
	$=$	$325$

Answer: Alice's row sum is $325$ . This is her combined score across all four tests.

row sum = 325

EXAMPLE 4 — NETWORK (CONNECTION) MATRIX

Three towns

P

Q

and

R

are joined by:

1

road

P

–

Q

2

roads

P

–

R

, and

1

road

Q

–

R

. Build the connection matrix.

SOLUTION

Order the rows and columns as $P$ , $Q$ , $R$ . The diagonal is 0 because there are no roads from a town to itself. Each off-diagonal entry holds the number of roads between those two towns.

$N = [\begin{matrix} 0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{matrix}]$

The matrix is symmetric ( $N_{13} = N_{31} = 2$ , and so on) because the roads are two-way.

Answer: the connection matrix $N$ is as shown above.

N = [\begin{matrix} 0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{matrix}]

Common pitfalls

Don't swap rows and columns mid-problem. Once you decide what rows and columns represent, stick with it for the whole question. If the question specifies which is which, follow that exactly.

One-way vs two-way networks. Two-way networks have symmetric matrices (

N_{i j} = N_{j i}

). One-way (directed) networks generally do not. Read the question carefully — "roads", "flights", and "messages" can each be one-way or two-way.

Row sum and column sum mean different things. "Alice's total" is a row sum; "Test 1 total across all students" is a column sum. Always say which sum and what it represents.

Transition matrices — columns sum to 1. A common error is to make the rows sum to 1 instead. In the convention used here, entry

(i, j)

is the proportion moving from category

j

(the column) to category

i

(the row), so each column must add up to 1.

Match the unit to the context. Entries can be counts (units sold), prices (dollars), proportions (percentages or decimals between 0 and 1), or scores. Always state units in your final answer.

Frequently asked questions

When should I use a matrix to represent data?

Use a matrix any time data falls naturally into a rectangular grid with two categories — for example, students by tests, stores by products, towns by towns. Each cell of the grid becomes one entry of the matrix.

How do I decide what goes in rows and columns?

Often the question specifies it directly: 'rows represent stores, columns represent products'. If not, pick whichever makes the most sense for the data and state your choice clearly. Just be consistent throughout the problem.

What does a row sum tell me?

A row sum is the total across all the categories in that row. For a test-score matrix with students as rows and tests as columns, a row sum gives a student's total across all the tests they sat.

What does a column sum tell me?

A column sum is the total down a single column. In the same test-score example, a column sum gives the total of all students' marks on one particular test.

What is a network or connection matrix?

A network matrix represents direct connections between points (such as roads between towns). Entry (i, j) is the number of direct connections from point i to point j. For two-way connections the matrix is symmetric; for one-way connections it isn't. Diagonal entries are usually zero — no connection from a point to itself.

What is a transition matrix?

A transition matrix describes how a quantity moves between categories over time — for example, customers switching brands. Entry (i, j) is the proportion that moves from category j to category i. Each column sums to one, since every member of category j must move somewhere (including staying).