Matrices · Form 5
Matrices: Revision Notes
A tight revision summary of Matrices for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
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.
Key ideas to revise
- 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.
One worked line for every key idea
- Order decides what is allowed. To ADD, both matrices must share an order: [[1,2],[3,4]] + [[5,6],[7,8]] works because both are 2×2, giving [[6,8],[10,12]]. To MULTIPLY, the columns of the first must equal the rows of the second, a 2×2 times a 2×1 is allowed and its answer is 2×1.
- Scalar multiplication touches every entry, not just the first: 3 × [[1,2],[-1,0]] = [[3,6],[-3,0]]. All four numbers are multiplied by 3.
- Matrix multiplication is row-into-column, then add: [[2,1],[0,3]] × [[4],[5]] = [[2·4 + 1·5],[0·4 + 3·5]] = [[13],[15]].
- Inverse of a 2×2: for M = [[4,3],[2,5]], determinant = 4·5 − 3·2 = 14, so M⁻¹ = (1/14)[[5,-3],[-2,4]]. Swap the diagonal, negate the other two, divide by 14.
Your pre-paper checklist for matrices
- Re-derive the inverse pattern by hand: swap the two leading-diagonal entries, negate the other two, then divide by the determinant. Drill it until it is automatic.
- Know what the paper gives you: the formula sheet prints the 2×2 inverse formula, so you never recall it blind, but you must substitute a, b, c, d in the right places.
- Always compute the determinant first. If it equals zero, stop, there is no inverse and no unique solution to the equations.
- For simultaneous equations, write them as AX = B and multiply B on the LEFT by A⁻¹, giving X = A⁻¹B, never BA⁻¹.
- The one habit that saves marks: substitute your final x and y back into both original equations to confirm they hold before you move on.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How can I check my inverse is right without a marking scheme?
Multiply your inverse by the original matrix. If the product comes out as the identity matrix [[1,0],[0,1]], your inverse is correct.
This works in either order, since a matrix and its inverse commute, and it takes under a minute, a reliable self-check when you have spare time at the end.
Do I have to memorise the inverse formula for the exam?
No, the 2×2 inverse formula is printed on the SPM Mathematics formula sheet, so you will not be asked to recall it blind. What you must be fluent in is reading off a, b, c, d from your matrix, computing the determinant, and placing the four entries in the swap-and-negate pattern correctly.
In what order should I revise the matrix operations?
Build up in layers: first order, addition and subtraction; then scalar multiplication; then matrix multiplication with the row-into-column rule; then the determinant; then the inverse; and finally using the inverse to solve simultaneous equations. Each step relies on the one before, so a firm early layer makes the harder parts feel routine.