Matrices questions look intimidating because they arrive as a paragraph of shopping, ticket sales or factory output, and only then turn into rows and columns. The good news is that almost every matrices word problem in O-Level E Math follows the same three moves: define your matrices, check the orders multiply, then say in words what your answer means.
A quick note before we start. Matrices sit in the E Math syllabus 4048, and different cohorts sit slightly different syllabus codes. Check with your school which topics your paper covers, then use this guide to drill the technique.
Why these questions trip students up
Students rarely lose marks on the arithmetic. They lose marks because:
- They multiply in the wrong order, so the orders do not match and the product does not exist.
- They set rows and columns up by instinct instead of by design.
- They compute the correct numbers, then write a vague interpretation like "the total" instead of "the total cost, in dollars, of the items bought on Monday".
All three are fixable with a routine.
Step 1: Label your rows and columns before you write any numbers
Before you write a single bracket, write two short lines in your working:
- Rows represent ...
- Columns represent ...
For example, if a stall sells three drinks over two days, decide: rows are days, columns are drink types. That gives a 2 by 3 sales matrix.
S = ( 20 15 10 ) Row 1 = Monday
( 25 12 18 ) Row 2 = Tuesday
Columns = drinks A, B, C
Prices belong in a column matrix so that the multiplication works:
P = ( 1.50 ) Drink A
( 2.00 ) Drink B
( 2.50 ) Drink C
Step 2: Use the order check to decide what multiplies what
This is the single most useful habit. Write the orders side by side:
(2 x 3)(3 x 1) gives a 2 x 1 matrix.
The inner numbers match, so the product exists, and the outer numbers tell you the shape of the answer. If you had tried P times S, you would have (3 x 1)(2 x 3), where the inner numbers 1 and 2 do not match, so the product is undefined. Matrix multiplication is not commutative, and examiners love to test whether you know that.
Step 3: Multiply, then read the answer back into English
SP = ( 20(1.50) + 15(2.00) + 10(2.50) ) = ( 85.00 )
( 25(1.50) + 12(2.00) + 18(2.50) ) ( 106.50 )
Now the interpretation, which is usually worth a mark on its own:
"The element 85 represents the total amount of money, in dollars, collected from drink sales on Monday."
Notice the three ingredients the mark scheme wants: what quantity, what units, and which day or category. Train yourself to include all three every time.
Totalling with a row of ones
If the question asks for the takings over both days, pre-multiply by a 1 by 2 matrix of ones:
(1 1)(2 x 1) gives (1 x 1), so
( 1 1 ) ( 85.00 ) = ( 191.50 )
( 106.50 )
The single element 191.50 is the total takings in dollars over Monday and Tuesday. Summing matrices of ones appear constantly, so recognise them on sight.
Profit questions
If cost prices are given as C, do the subtraction first inside a bracket:
Profit matrix = S(P - C).
With C = (0.90, 1.20, 1.60) as a column, P - C = (0.60, 0.80, 0.90), and
Monday: 20(0.60) + 15(0.80) + 10(0.90) = 33.00
Tuesday: 25(0.60) + 12(0.80) + 18(0.90) = 40.80
So the total profit is $73.80.
Percentage changes
A 10 percent price increase is scalar multiplication: use 1.1P, not P + 10. A 15 percent discount is 0.85P. State the scalar clearly in your working so the method mark is safe even if you slip in the arithmetic.
Common mistakes to check in the last minute
- Undefined products. Always write the orders before multiplying.
- Row times column, not column times row. Go across the row of the first matrix and down the column of the second.
- Missing units. Money answers need dollars and two decimal places.
- Vague interpretations. Name the row and the column your element came from.
- Rounding too early. Keep exact values inside the matrix and round only at the final answer.
A five-minute drill you can repeat
Take any past-year matrices question, cover the numbers, and just write the orders and the labels. If you can predict that the answer is a 2 by 1 matrix representing daily totals before doing any arithmetic, you have understood the question. Then do the arithmetic. This separates the thinking marks from the calculator marks, which is exactly how the paper is graded.
If matrices, or any other E Math topic, still feels like guesswork under exam conditions, our tutors sat these same papers recently and can walk you through the setup line by line. Message us on WhatsApp at https://wa.link/dsgbkf to book a free trial class.