Matrices · Form 5
Matrices: Paper 2 Answering Guide
How Matrices appears in Mathematics Paper 2 (Subjective), and how to lay out your working so you earn every method mark.
How it is examined
Paper 2 typically asks for a product of matrices, then an inverse, then to use that inverse to solve a pair of simultaneous equations. Marks are lost to arithmetic slips and to multiplying in the wrong order, so neat, labelled working matters.
Showing your working
Mathematics Paper 2 (Subjective) is worth 100 marks and gives marks for the steps, not only the answer. Write each line clearly: state the formula or rule, substitute the numbers, then simplify.
If the question carries units, carry them through to the final line.
Worked Paper 2 example: inverse then simultaneous equations
Question: The matrix P = [[2,3],[1,4]]. (a) Find the determinant of P.
(b) Hence find P⁻¹. (c) Using P⁻¹, solve the simultaneous equations 2x + 3y = 7 and x + 4y = 6.
- (a) State the rule: determinant = ad − bc. Substitute a = 2, b = 3, c = 1, d = 4: det P = (2)(4) − (3)(1) = 8 − 3 = 5.
- (b) State the inverse rule: P⁻¹ = (1/det)[[d,-b],[-c,a]]. Substitute: P⁻¹ = (1/5)[[4,-3],[-1,2]].
- (c) Write the system in matrix form: [[2,3],[1,4]][[x],[y]] = [[7],[6]], i.e. PX = B.
- (c) State the solving rule: multiply on the left by P⁻¹, so X = P⁻¹B = (1/5)[[4,-3],[-1,2]][[7],[6]].
- (c) Multiply row into column: top = 4·7 + (−3)·6 = 28 − 18 = 10; bottom = (−1)·7 + 2·6 = −7 + 12 = 5. So X = (1/5)[[10],[5]].
- (c) Simplify: X = [[10/5],[5/5]] = [[2],[1]]. Answer: x = 2, y = 1.
- Check: 2(2) + 3(1) = 7 and (2) + 4(1) = 6, both original equations hold.
Where the method marks are won and lost
The examiner is marking your layout as much as your answer. The two lines PX = B and X = P⁻¹B each declare that you know the method, and the row-into-column line shows the multiplication was done, not guessed.
A correct x = 2, y = 1 with none of that scaffolding often scores far below full marks, because there is nothing on the page for the method marks to attach to.
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
How do I know the exam wants the inverse method rather than elimination?
Read the linking words. When part (c) says 'Hence, using P⁻¹, solve...', the word 'hence' ties it to the inverse you just found, so you must use matrices.
If a standalone question simply says 'solve', elimination is allowed, but when earlier parts built the inverse, the matrix method is clearly what earns the marks.
Do I earn marks just for writing PX = B before solving?
Yes. Setting the equations up as PX = B and then stating X = P⁻¹B are recognised method steps that carry marks in the scheme.
They cost seconds to write and protect you: even if an arithmetic slip spoils the final numbers, these lines can still secure the method marks for that part.
Can I leave my answer as a fraction, or must it be a whole number?
Leave it in whatever exact form the arithmetic gives. Here it simplified to whole numbers, x = 2 and y = 1, so present those.
If a different system gives x = 21/11, keep the exact fraction rather than rounding, the mark is for an exact value unless the question explicitly asks for a decimal to a stated number of places.