Form 5 · Relationship and Algebra
Matrices
Matrices are rectangles of numbers with their own arithmetic, a compact way to handle a lot of data at once.
What is Matrices?
A matrix is an array of numbers arranged in rows and columns. This chapter teaches the basic operations, addition, subtraction, multiplication by a number and by another matrix, and the inverse of a 2×2 matrix, which is used to solve simultaneous equations.
Content standards (DSKP)
The DSKP KSSM sets these content standards for this chapter:
- 2.1 Matrices
- 2.2 Basic Operation on Matrices
The key ideas
Order and when operations are allowed
You can only add matrices of the same order, and multiply only when the columns of the first match the rows of the second.
Matrix multiplication
Multiply rows into columns and add, the one operation students most often set out wrongly.
The inverse of a 2×2 matrix
The given formula uses the determinant ad − bc; if it is zero there is no inverse. The inverse solves simultaneous equations neatly.
How this chapter 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.
Formulae given in the exam for this chapter
Common mistakes to avoid
- Multiplying matrices in the wrong order (order matters)
- Sign errors in the inverse formula
- Trying to invert a matrix whose determinant is zero
The identity matrix and what an inverse really means
For ordinary numbers, the inverse of 5 is 1/5 because 5 × 1/5 = 1, and 1 is the number that changes nothing when you multiply by it. Matrices have the same idea.
The identity matrix of order 2 is I = [[1, 0], [0, 1]]; multiplying any 2×2 matrix by I leaves it unchanged, so I plays the role that 1 plays for numbers. The inverse of a matrix A, written A⁻¹, is the matrix that satisfies A × A⁻¹ = I.
That is why the inverse is so useful: multiplying by A⁻¹ 'undoes' multiplying by A, just as dividing by 5 undoes multiplying by 5. Keeping this picture in mind explains what you are aiming for when you compute an inverse, and why only square matrices can have one, you need the same number of rows and columns for A × A⁻¹ to give the square identity.
Solving simultaneous equations the matrix way
The reason the inverse matters in Paper 2 is that it solves a pair of simultaneous equations in one clean move. Take two equations such as 3x + y = 7 and 5x + 2y = 12.
Write them as a single matrix equation MX = C, where M is the matrix of coefficients [[3, 1], [5, 2]], X is the column [x, y] of unknowns, and C is the column [7, 12] of the right-hand sides. To solve for X, multiply both sides on the left by M⁻¹: since M⁻¹M = I, you get X = M⁻¹C.
So the whole method is: build M, C and X; find M⁻¹; then multiply M⁻¹ by C to read off x and y. The order matters, the inverse must go on the left of C, never the right, because matrix multiplication is not commutative.
This is the standard structured question, and knowing the MX = C setup means you are never unsure what to do next.
The determinant: your check before inverting
Before you invert a 2×2 matrix [[a, b], [c, d]], compute its determinant, ad − bc. The inverse formula printed on the exam formula sheet is A⁻¹ = (1 / (ad − bc)) × [[d, −b], [−c, a]], and the determinant is the number you divide by.
If the determinant is any non-zero value the inverse exists and you carry on; if it comes out as zero the matrix is singular and has no inverse at all, because you cannot divide by zero. So the determinant is both a green light and a warning: work it out first, and only proceed if it is non-zero.
In SPM a determinant of zero almost always means an arithmetic slip in the multiplication ad or bc, so recheck those products before concluding the matrix is genuinely singular. Watch the order too, it is ad − bc, not bc − ad or ad + bc, since getting that wrong flips the sign of the whole inverse.
A worked exam-style example
This example runs the full Paper 2 chain: a determinant, an inverse, and using that inverse to solve simultaneous equations.
- (a) With a = 3, b = 1, c = 5, d = 2, the determinant is ad − bc = (3)(2) − (1)(5) = 6 − 5 = 1.
- (b) The inverse formula gives M⁻¹ = (1 / (ad − bc)) × [[d, −b], [−c, a]] = (1/1) × [[2, −1], [−5, 3]].
- So M⁻¹ = [[2, −1], [−5, 3]].
- (c) Write the equations as M[x, y] = [7, 12], so [x, y] = M⁻¹[7, 12] = [[2, −1], [−5, 3]] × [7, 12].
- Top row: x = (2)(7) + (−1)(12) = 14 − 12 = 2.
- Bottom row: y = (−5)(7) + (3)(12) = −35 + 36 = 1.
- So x = 2 and y = 1.
- Check in the original equations: 3(2) + 1 = 7 ✓ and 5(2) + 2(1) = 10 + 2 = 12 ✓.
How to study this chapter
Study Matrices
Formulas
Frequently asked questions
How this chapter is examined
SPM Mathematics assesses this chapter across Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective), drawing on the DSKP content standards above. Paper 2 gives marks for working, so showing every step matters.
Common mistakes to avoid
Multiplying matrices in the wrong order (order matters); Sign errors in the inverse formula; Trying to invert a matrix whose determinant is zero.
Formulae given in the exam for this chapter
Yes, Inverse of a 2x2 matrix appear on the formula sheet the exam provides. Anything else in this chapter you are expected to know.
When can I add two matrices, and when can I multiply them?
Add or subtract only matrices of the same order, the same rows and columns, element by element. To multiply, the number of columns in the first matrix must equal the number of rows in the second.
So a 2×2 times a 2×1 works and gives a 2×1, but a 2×1 times a 2×2 does not.
Why can't I just swap the order when multiplying?
Because matrix multiplication uses rows of the first against columns of the second, AB and BA combine different numbers and usually give different results, sometimes BA is not even defined. Always keep the order the question sets.
When solving equations, the inverse goes on the left: X = M⁻¹C, never CM⁻¹.
What if the determinant comes out as zero?
Then the matrix has no inverse, it is called singular, and the inverse method cannot be used, since you would be dividing by zero. In SPM a zero determinant usually points to an arithmetic slip, so recheck ad − bc first.
If it is genuinely zero, the equations have no unique solution.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)· SPM: Format Pentaksiran mulai 2021, Matematik (1449)
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