Year 11 General Matrices And Matrix Arithmetic

Problem-Solving And Modelling With Matrices

20 practice questions 2 video lessons Theory + worked examples

Theory

Real-world problems translate into matrices, then the right operation: addition for combining data, subtraction for change, scalar multiplication for percentage changes, and matrix multiplication for quantity × price calculations. Always state units and context in the final answer.

Real-world problems involving matrices share a common pattern: translate the situation into matrices, choose the right operation, then interpret the result back in context. The four basic operations each have a typical use.

The most powerful pattern is quantity $\times$ price. If $Q$ is a matrix of quantities (rows = stores or buyers, columns = products) and $P$ is a column matrix of unit prices, then $Q P$ gives a column of total revenues — one entry per row of $Q$ . The "products" dimension cancels through the multiplication.

Many problems chain operations — combine, then multiply, then scale. Work through one step at a time and keep track of orders at each stage.

The first diagram is a decision reference — match the real-world situation to the matrix operation. The second visualises the powerful quantity × price pattern, showing how the shared dimension cancels.

Each common situation maps to one of four matrix operations.

Q

is stores × products,

P

is products × 1, so

Q P

is stores × 1: one revenue per store.

There are no new formulas — just operation templates for the common modelling contexts.

Quantity × Price = Revenue

$\underset{stores \times products}{\underset{⏟}{Q}} \cdot \underset{products \times 1}{\underset{⏟}{P}} = \underset{stores \times 1}{\underset{⏟}{Q P}}$

Q P = revenue per store

Percentage-change scalars

Change	Multiply by
10% increase	$1.10$
20% increase	$1.20$
10% GST added	$1.10$
15% discount	$0.85$
25% off (sale)	$0.75$

Order matters for matrix products. QP and PQ are different operations, and one may be undefined. Choose the order that makes contextual sense — the rows of the first matrix should be the categories you want in the answer.

Translating a word problem to matrices

Identify the unknowns and what categories the data sits in.
Build the matrices — decide what each row and column represents and write the numbers in.
Choose the operation using the decision table: combining data is $+$ , change is $-$ , percentage is scalar, quantity × price is a product.
Compute the operation.
State the answer in context with units. " $$ 925$ total revenue", not just "925".

Setting up a $Q P$ revenue calculation

Quantity matrix $Q$ : rows are buyers (or stores), columns are products. Entry is units.
Price column $P$ : one row per product, with the unit price as the entry.
Form $Q P$ . The result is a column with one revenue per row of $Q$ .
For a grand total, sum the entries of $Q P$ .

Multi-step problems

Work through one operation at a time. Label intermediates clearly (e.g. "new prices").
Check the order after each step — operations may change the order of the matrix.
Always finish with a sentence in plain English describing the result in context.

EXAMPLE 1 — QUANTITY × PRICE (SINGLE BUYER)

A customer buys

2

loaves of bread (

$ 4

each),

3

cartons of milk (

$ 5

each) and

1

dozen eggs (

$ 6

). Use a matrix product to find the total cost.

SOLUTION

Let $P = [\begin{matrix} 4 & 5 & 6 \end{matrix}]$ be a row matrix of prices and $Q = [\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}]$ be a column matrix of quantities. $P$ is $1 \times 3$ , $Q$ is $3 \times 1$ , so $P Q$ is $1 \times 1$ .

$P Q$	$=$	$(4) (2) + (5) (3) + (6) (1)$
	$=$	$8 + 15 + 6$
	$=$	$$ 29$

Answer: the customer pays $$ 29$ in total.

P Q = $ 29

EXAMPLE 2 — COMBINING TWO PERIODS

Cafe sales for Week 1 and Week 2 (rows = stores, columns = coffee, tea) are

W_{1} = [\begin{matrix} 50 & 30 \\ 40 & 25 \end{matrix}]

and

W_{2} = [\begin{matrix} 60 & 35 \\ 50 & 30 \end{matrix}]

. Find the two-week total.

SOLUTION

Combining two snapshots → add the matrices entry-by-entry.

$T$	$=$	$W_{1} + W_{2}$
	$=$	$[\begin{matrix} 110 & 65 \\ 90 & 55 \end{matrix}]$

Answer: the two-week total is shown above. For example, Store 1 sold $110$ coffees and $65$ teas.

T = [\begin{matrix} 110 & 65 \\ 90 & 55 \end{matrix}]

EXAMPLE 3 — REVENUE PER DAY

A cafe records cups sold each day (rows = days, columns = coffee, tea):

S = [\begin{matrix} 40 & 25 \\ 50 & 30 \\ 35 & 20 \end{matrix}]

. Coffee costs

$ 5

and tea costs

$ 4

. Find the revenue per day and the total.

SOLUTION

Let $P = [\begin{matrix} 5 \\ 4 \end{matrix}]$ . $S$ is $3 \times 2$ and $P$ is $2 \times 1$ , so $S P$ is $3 \times 1$ — one revenue per day.

Day 1	$=$	$40 (5) + 25 (4) = 300$
Day 2	$=$	$50 (5) + 30 (4) = 370$
Day 3	$=$	$35 (5) + 20 (4) = 255$

$S P = [\begin{matrix} 300 \\ 370 \\ 255 \end{matrix}] AUD$

Grand total: $$ 300 + $ 370 + $ 255 = $ 925$ .

Answer: daily revenues are $$ 300$ , $$ 370$ , and $$ 255$ ; the three-day total is $$ 925$ .

total = $ 925

EXAMPLE 4 — TWO OPERATIONS (PRICE RISE THEN PURCHASE)

A store's prices are

P = [\begin{matrix} 10 & 5 & 8 \end{matrix}]

AUD. Prices rise by

20 %

. A customer buys

Q = [\begin{matrix} 3 \\ 7 \\ 4 \end{matrix}]

. Find the new total cost.

SOLUTION

Step 1 — apply the price rise. Multiply $P$ by the scalar $1.20$ .

$1.20 P = [\begin{matrix} 12 & 6 & 9.60 \end{matrix}]$

Step 2 — multiply by the quantities. The new prices are $1 \times 3$ ; $Q$ is $3 \times 1$ ; the product is $1 \times 1$ .

Cost	$=$	$(1.20 P) Q$
	$=$	$12 (3) + 6 (7) + 9.60 (4)$
	$=$	$36 + 42 + 38.40$
	$=$	$$ 116.40$

Answer: the total at the new prices is $$ 116.40$ .

cost = $ 116.40

Common pitfalls

Matrix multiplication is not commutative.

Q P

and

P Q

are different operations, and often only one of them is defined. Choose the order that makes contextual sense — the rows of the first matrix should be the categories you want as rows of the answer.

Use the right percentage factor. A

15 %

discount means multiplying by

0.85

, not

0.15

. A

20 %

increase means multiplying by

1.20

. A

10 %

GST added means multiplying by

1.10

State units and context. "925" by itself isn't a complete answer. Write "

$ 925

total revenue across the three days" — much more useful and almost always required for full marks.

Add only when the two matrices have the same order. Combining Week 1 and Week 2 needs both to have the same rows and columns. If they don't, you can't simply add.

For multi-step problems, label intermediates. Don't combine three operations in one giant expression. Compute "new prices = 1.20 P", then "cost = (new prices) Q". Each step is its own labelled matrix.

Frequently asked questions

When should I add two matrices instead of multiplying?

Add when you are combining two snapshots of the same kind of data — Week 1 sales plus Week 2 sales, opening stock plus arrivals. Multiply when one matrix is quantities and the other is prices, or when applying a transition rule.

When should I use scalar multiplication?

Use scalar multiplication when EVERY entry should change by the same factor — a 10 percent price rise (multiply by 1.10), a 15 percent discount (multiply by 0.85), or a currency conversion (multiply by the exchange rate).

How does quantity times price work as a matrix product?

If Q is a matrix of quantities (rows are stores or buyers, columns are products) and P is a column matrix of unit prices, then Q times P gives a column of total revenues — one entry per row of Q. The products dimension cancels in the multiplication.

Does the order of multiplication matter for a real-world problem?

Yes — matrix multiplication is not commutative. QP and PQ are different operations and often only one of them is defined. The choice depends on which dimension you want to keep in the answer.

How do I apply a 20 percent increase using matrices?

Multiply the price matrix by the scalar 1.20. This applies the same 20 percent increase to every entry at once. For a 15 percent discount, multiply by 0.85 — and for a 10 percent GST addition, multiply by 1.10.

How do I combine two matrix operations in one problem?

Do them in order, keeping track of the orders at each step. For example: apply the scalar first to get new prices, then multiply by the quantity matrix to get the total. Compute step by step and label each intermediate result clearly.

Video Lessons

Practice Questions

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Problem-Solving And Modelling With Matrices

📖 Theory

Quantity × Price = Revenue

Percentage-change scalars

Translating a word problem to matrices

Setting up a QP revenue calculation

Multi-step problems

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

Setting up a $Q P$ revenue calculation

Video Lessons

Practice Questions