RECIPEThe 3 legal moves (E.R.O)
Scale
Ri → kRimultiply a row by k ≠ 0
Add
Ri → Ri + kRjadd k × another row
Exam recipe: "Find E₁ and E₂ that reduce P to Q"
- Compare P and Q row by row. Which rows are unchanged? Did two rows swap? Is a row a multiple (×2, ×−⅓)? Is a row = itself + another row?
- Write the operations in order: first one → E₁, second one → E₂ (because Q = E₂E₁P, E₁ acts first).
- Do each operation on the identity (I₃ for 3 rows, I₂ for 2 rows!). That is E.
- Inverses = undo: swap → same swap · ×k → ×1/k · +kRj → −kRj. Do the undo move on I.
Senang ceritaAn elementary matrix is just "I with one move done to it". Its inverse is "I with the opposite move done to it".
PLAY ITBuild E and E⁻¹
WATCH ITJul 2025 Q2(a)
DO ITPractice
REMEMBERMemory hooks
🔁Swap is its own undo. E₁⁻¹ = E₁ for a swap.
TRAPIf P has 2 rows (like Jul 2024), E is 2×2, made from I₂, not I₃.
PROVE IT3 quick checks
RECIPESpot the staircase
REF (row echelon form)
1 2 -1 | 3;0 1 4 | 2;0 0 1 | -1- Zero rows at the bottom
- First non-zero number of each row = 1 (leading 1)
- Each leading 1 is to the right of the one above
RREF (reduced)
1 0 0 | 5;0 1 0 | -2;0 0 1 | 1- It is REF, plus…
- every column with a leading 1 has zeros everywhere else (above too)
Senang ceritaREF = zeros below the stairs. RREF = zeros below AND above each leading 1.
PLAY ITRow-operation sandbox
Use the 3 moves to turn this into REF, then RREF. Every move is written out entry by entry, exactly how you write it in the exam.
DO ITREF, RREF or neither?
REMEMBERGood habits
✍️Write each move next to the arrow, e.g. R2 → R2 − 2R1. Marks are given for showing them.
1️⃣Want a 1 on top? Swap a row that already starts with 1 to the top. It saves fractions.
PROVE IT3 quick checks
RECIPESolve any system
- Augmented matrix (A | b): coefficients on the left, answers on the right. Missing variable = 0.
- Column 1: get a leading 1 on top (swap or scale), then make everything below it 0 (Ri → Ri − kR1).
- Column 2: leading 1 in row 2, zeros below. Then column 3…
- Gauss: stop at REF, write equations, back-substitute from the bottom up. Gauss-Jordan: also make zeros ABOVE each leading 1 (RREF), then read the answer off directly.
- Read the result: one answer = unique · a row 0 = 0 = free variable → infinitely many · a row 0 = non-zero → no solution.
DRAG ITWhat "solution" means: lines meeting
meet once → unique
parallel → none
same line → infinite
WATCH ITJan 2023 Q2(b) · Gauss-Jordan
CALCULATORCheck a 3-unknown answer
ClassWizMENU A (Equation/Func) →
1 Simul Equation →
3 unknowns → type a, b, c, d of each row,
= after each →
= shows x, y, z.
ES PLUSMODE 5 (EQN) →
2 (anX + bnY + cnZ = dn) → type numbers →
=
This only works for unique solutions. With free variables it says "Infinite Sol" or "No Solution". Even that tells you which case you have!
DO ITPractice (unique solutions)
PROVE IT3 quick checks
RECIPELeading vs free
- Row-reduce to REF (Gauss) or RREF (Gauss-Jordan).
- Circle the leading 1s. Their columns are the leading variables.
- Every other variable is free: let it equal a parameter (x₂ = s, x₄ = t, …).
- Move the free variables to the right side and solve for each leading variable (from the bottom row up).
- Write ALL variables in the final answer, including "x₂ = s, x₄ = t", and add "s, t ∈ ℝ".
Senang ceritaLeading = the ones you solve for. Free = the ones that can be anything.
WATCH ITJul 2025 Q2(b) · Gaussian elimination
DO ITPractice
TRAPForgetting to write the free variable itself (x₄ = t) in the final answer loses a mark.
PROVE IT3 quick checks
RECIPEThe last-row test
- Write the augmented matrix and row-reduce until the last row is (0 0 coefficient | right side).
- Factorise the coefficient, e.g. k² − 4 = (k − 2)(k + 2).
- Unique: coefficient ≠ 0 → k ≠ ±2.
- Infinitely many: coefficient = 0 AND right side = 0 → the row reads 0 = 0.
- No solution: coefficient = 0 but right side ≠ 0 → the row reads 0 = something. Impossible!
Senang cerita"Zero-zero, infinite hero. Zero-something, got nothing."
SLIDE ITWatch k change the answer
WATCH ITJul 2024 Q2(c), every line
DO ITPractice (endless, exam style)
REMEMBERTwo-letter versions (α and β)
Feb 2025: last row (α² − 9)z = 2β. Unique: α ≠ ±3 · Infinite: α = ±3 AND β = 0 · None: α = ±3 AND β ≠ 0.
Jan 2023: last row (α + 3)z = β² − 16. Unique: α ≠ −3 · Infinite: α = −3, β = ±4 · None: α = −3, β ≠ ±4.
TRAPFor "unique" write k ≠ 2 and k ≠ −2 (or k ≠ ±2), not just one of them.
PROVE IT3 quick checks
RECIPE 1Homogeneous (b = 0)
- Homogeneous means every right side is 0. Then x = y = z = 0 always works: the trivial solution.
- So a homogeneous system is ALWAYS consistent. It can never have "no solution".
- Same last-row test with right side 0: coefficient ≠ 0 → only trivial. Coefficient = 0 → 0 = 0 → non-trivial (infinitely many) solutions.
Example (Feb 2026)(ρ² − 1)z = 0 → trivial only when ρ ≠ ±1, non-trivial when ρ = ±1. Both consistent because x = y = z = 0 always works.
RECIPE 2A⁻¹ by elementary row operations
- Write the big matrix [A | I] (A on the left, identity on the right).
- Row-reduce the LEFT side to I (Gauss-Jordan). Every move changes the WHOLE row, both sides.
- When the left is I, the right side is A⁻¹: [A | I] → [I | A⁻¹].
- If a row of zeros appears on the left → A has no inverse.
- Bonus: solving AX = b by inverse → X = A⁻¹b.
WATCH ITJul 2024 Q2(b)
CALCULATORCheck A⁻¹
ClassWiz: OPTN MatA x⁻¹ = · ES PLUS: SHIFT43 x⁻¹ =. The question says "by elementary row operations", so the [A | I] working must be on paper.
DO ITPractice
PROVE IT3 quick checks
How to scoreQ2 is the biggest question. For (c), the marker wants: the reduced matrix (2m) + each case with its reason (e.g. "0 = 4, inconsistent"). Always write the last row as an equation before answering.