Matrices · 2.2.7
Solving problems with matrices
This standard brings together every matrix skill, addition, multiplication, the identity matrix, the inverse matrix, and the matrix method, to solve real or applied problems. Students typically translate a situation into equations, form a matrix equation, find the inverse, solve for the unknowns, and interpret the answer in context.
The official learning standard (2.2.7)
“Solve problems involving matrices. (Exploratory activities involving various types of matrix multiplication, including multiplication with the identity matrix, should be carried out; the symbol I for the identity matrix should be introduced.
Exploratory activities involving AA⁻¹ = A⁻¹A = I should also be carried out; the symbol A⁻¹ and the term determinant should be introduced, along with the condition for an inverse matrix to exist.)”
What it means
This standard brings together every matrix skill, addition, multiplication, the identity matrix, the inverse matrix, and the matrix method, to solve real or applied problems. Students typically translate a situation into equations, form a matrix equation, find the inverse, solve for the unknowns, and interpret the answer in context.
How it is examined
This appears in Paper 2 as a longer word-problem question, often set in a shopping or cost context with two unknown quantities. Students must set up two equations from the given information, convert them to matrix form, find the inverse, and state the final answer with correct units.
Worked example
A stationery shop sells pens and pencils. Puan Lee buys 2 pens and 1 pencil for RM8, while her son buys 1 pen and 1 pencil for RM5.
Using the matrix method, find the price of one pen and one pencil.
- Let the price of a pen be RM p and a pencil be RM c. Form the equations: 2p + c = 8 and p + c = 5.
- Write in matrix form: [[2, 1], [1, 1]] [[p], [c]] = [[8], [5]]
- Find the determinant of A: (2×1) − (1×1) = 2 − 1 = 1
- Find A⁻¹ = 1/1 × [[1, −1], [−1, 2]] = [[1, −1], [−1, 2]]
- Compute X = A⁻¹B: p = (1×8) + (−1×5) = 8 − 5 = 3
- c = (−1×8) + (2×5) = −8 + 10 = 2
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I know a problem should be solved using matrices?
Look for a situation with two unknown quantities and two pieces of information that each connect them, such as two purchases or two mixtures. If you can write two linear equations in two unknowns, the matrix method is a valid way to solve them.
Do I need to verify AA⁻¹ = I in every problem?
It is not always required if the question only asks for the final values, but showing this check demonstrates understanding and can catch an error in your inverse matrix before it affects your final answer, so it is good practice when time permits.
What should I do if the determinant turns out to be zero in a word problem?
A zero determinant means the two equations do not give a unique solution, so re-check that you copied the numbers correctly from the question. In a genuine SPM problem the determinant is designed to be non-zero, so a zero result usually signals a transcription mistake.