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Matrices · Form 5

Matrices: Worked Examples (KBAT)

Real situations where you must spot the matrix method yourself: pricing from two orders, totalling costs across branches, and scaling weekly production. These stretch students to translate a story into matrices and back.

Written by the spmmath.com.my editorial team.

Worked example 1

A stall sells nasi lemak and teh tarik at fixed prices. One customer pays RM13 for 3 nasi lemak and 2 teh tarik.

Another pays RM22 for 4 nasi lemak and 5 teh tarik. Using a matrix method, find the price of one nasi lemak and one teh tarik.

  1. Let x = price of one nasi lemak, y = price of one teh tarik (in RM). Then 3x + 2y = 13 and 4x + 5y = 22.
  2. Matrix form: [[3, 2], [4, 5]] [[x], [y]] = [[13], [22]].
  3. Determinant = (3×5) − (2×4) = 15 − 8 = 7.
  4. Inverse = (1/7)[[5, −2], [−4, 3]]; multiply by [[13], [22]]: [[x], [y]] = (1/7)[[(5×13)+(−2×22)], [(−4×13)+(3×22)]].
  5. Simplify: (1/7)[[65 − 44], [−52 + 66]] = (1/7)[[21], [14]] = [[3], [2]].

Worked example 2

A uniform supplier charges RM25 per shirt and RM35 per pair of trousers. Shop P orders 40 shirts and 30 trousers; Shop Q orders 25 shirts and 50 trousers.

Represent the orders as one matrix and the prices as another, then use matrix multiplication to find each shop's total bill.

  1. Quantity matrix Q = [[40, 30], [25, 50]] (rows = Shop P, Shop Q; columns = shirts, trousers). Price matrix P = [[25], [35]].
  2. Total = Q × P. Shop P: (40×25) + (30×35) = 1000 + 1050 = 2050.
  3. Shop Q: (25×25) + (50×35) = 625 + 1750 = 2375.
  4. So Q × P = [[2050], [2375]].

Worked example 3

A bakery has two branches. On each weekday, output (cakes, breads) is given by W = [[30, 45], [20, 50]] (row 1 = Branch A, row 2 = Branch B).

On each weekend day the bakery produces 1.5 times the weekday amount. Using matrix operations, find the total production of each item at each branch over one full week (5 weekdays + 2 weekend days).

  1. Weekend daily output = 1.5 × W (scalar multiplication).
  2. Week total = 5 weekdays + 2 weekend days = 5W + 2(1.5W) = 5W + 3W = 8W.
  3. Compute 8W = 8 × [[30, 45], [20, 50]] = [[240, 360], [160, 400]].

Worked example 4

At a stationery shop, 2 pens and 3 files cost RM16, while 5 pens and 2 files cost RM18. Using the matrix method, find the price of a pen and the price of a file.

  1. Let a pen cost RM x and a file cost RM y: 2x + 3y = 16 and 5x + 2y = 18.
  2. Matrix form: [[2, 3], [5, 2]] [[x], [y]] = [[16], [18]].
  3. Determinant = (2 × 2) − (3 × 5) = 4 − 15 = −11.
  4. [[x], [y]] = (1/−11) [[2, −3], [−5, 2]] [[16], [18]] = (1/−11) [[−22], [−44]] = [[2], [4]].

Worked example 5

A caterer makes two set meals. Set A uses 2 pieces of chicken and 3 scoops of rice; Set B uses 1 piece of chicken and 4 scoops of rice.

On one day the caterer makes 50 Set A and 80 Set B. Chicken costs RM3.50 per piece and rice costs RM0.80 per scoop.

Use matrix multiplication to find the total cost of the chicken and rice.

  1. Meals × ingredients: [50, 80] × [[2, 3], [1, 4]] = [50×2 + 80×1, 50×3 + 80×4] = [180, 470] (total pieces of chicken and scoops of rice).
  2. Ingredients × unit cost: [180, 470] × [[3.50], [0.80]] = 180 × 3.50 + 470 × 0.80.
  3. Calculate: 630 + 376 = 1006.

Worked example 6

A tuition centre has two branches. The number of students is S = [[40, 25], [30, 35]], where the rows are Branch A and Branch B and the columns are Form 4 and Form 5.

Monthly fees are RM120 for Form 4 and RM150 for Form 5. Use matrix multiplication to find each branch's monthly collection, then find the total collection the next month if fees rise by 10% with the same students.

  1. Multiply S by the fee column: [[40, 25], [30, 35]] [[120], [150]].
  2. Branch A: 40×120 + 25×150 = 4800 + 3750 = 8550.
  3. Branch B: 30×120 + 35×150 = 3600 + 5250 = 8850.
  4. This month's total = 8550 + 8850 = 17400.
  5. Next month (+10%): 17400 × 1.10 = 19140.

Worked example 7

A school sports day awards points for medals: gold = 5 points, silver = 3 points, bronze = 1 point. The medal table for three houses is: House Red, 4 gold, 3 silver, 2 bronze; House Blue, 3 gold, 5 silver, 2 bronze; House Green, 2 gold, 4 silver, 6 bronze.

Using matrix multiplication, find the total points scored by each house and state which house is the champion.

  1. Write the medal matrix (rows = houses, columns = gold, silver, bronze): M = [[4, 3, 2], [3, 5, 2], [2, 4, 6]].
  2. Write the points matrix as a column: P = [5, 3, 1].
  3. Multiply M by P: House Red = (4×5) + (3×3) + (2×1) = 20 + 9 + 2 = 31.
  4. House Blue = (3×5) + (5×3) + (2×1) = 15 + 15 + 2 = 32.
  5. House Green = (2×5) + (4×3) + (6×1) = 10 + 12 + 6 = 28.
  6. Compare: 32 > 31 > 28, so House Blue is the champion with 32 points.

Worked example 8

A stall sells nasi lemak and teh tarik at fixed prices. One customer pays RM13 for 3 nasi lemak and 2 teh tarik.

Another pays RM22 for 4 nasi lemak and 5 teh tarik. Using a matrix method, find the price of one nasi lemak and one teh tarik.

  1. Let a nasi lemak cost RM x and a teh tarik cost RM y. Write the equations: 3x + 2y = 13 and 4x + 5y = 22.
  2. Write in matrix form: [[3, 2], [4, 5]] [x, y] = [13, 22].
  3. Find the determinant: (3×5) − (2×4) = 15 − 8 = 7.
  4. Find the inverse: (1/7)[[5, −2], [−4, 3]].
  5. x = [(5×13) + (−2×22)] ÷ 7 = (65 − 44) ÷ 7 = 21 ÷ 7 = RM3.00.
  6. y = [(−4×13) + (3×22)] ÷ 7 = (−52 + 66) ÷ 7 = 14 ÷ 7 = RM2.00.

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

What makes a matrices KBAT question different from a standard question?

KBAT matrix questions embed the maths in a real-life context, like stock inventories, cost comparisons, or coding, and require you to set up the matrices yourself from a word problem before doing several linked operations. The challenge is usually translating the scenario into correct matrices, not the arithmetic itself.

How should students plan their working for a multi-step matrix KBAT question?

Break the problem into stages: identify what each matrix represents, decide which operation (addition, multiplication, or inverse) links the stages, and label matrices clearly as you go. Writing a short plan before calculating helps avoid combining matrices in the wrong order partway through a long question.

What earns the higher-order thinking marks in these questions?

Marks reward correct interpretation of the context into matrix form, logical sequencing of operations, and a final answer related back to the original question, for example stating what a total matrix means in terms of cost or quantity. Pure calculation without context interpretation will not earn full KBAT credit.

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