MAT423 · linear algebra I
WELCOME TO
Amirrul Mukminin's personal
MAT423 interactive notes
Specifically for the test: Chapters 1–3. Read the recipe, drag it, then do it yourself.
★ Test: Friday 13 November, 3–5 pm
PICK A CHAPTER
1
Matrices
Size & types, add/multiply, transpose, 2×2 inverse, find C in CAB = M, letter questions
★ Q1 · ≈11m2
Systems of linear equations
Elementary matrices, REF/RREF, Gauss & Gauss-Jordan, free variables, find k, A⁻¹ by ERO
★ Q2 · ≈32m3
Determinants
Cofactors, singular matrices, row/column operations, |cA| = cⁿ|A|, adjoint inverse, Cramer's rule
★ Q3 · ≈17m★
Mock test
A real past paper (Q1–Q3) with a 2-hour timer. Write, then reveal the model answers and mark yourself.
What the paper asks
From 5 past papers (Feb 2026, Jul 2025, Feb 2025, Jul 2024, Jan 2023). The pattern never changes, only the numbers.
Q1(a) A⁻¹ or Aᵀ · find C in CAB = M · (b) letters: (MN)ᵀ, (DE)⁻¹Ch 1
Q2(a) E₁, E₂ and inverses · (b) Gauss / Gauss-Jordan · (c) find k: unique / none / infiniteCh 2
Q3(a) cofactor / singular / letter det · (b) |2A⁻¹BᵀC| · (c) cofactor matrix + adjoint, CramerCh 3
Q4–5Linear transformations, eigenvaluesNot in this test
Tonight's plan
1 · 45 min
Ch 2 firstBiggest marks. Recipe 5 (find k) is in every paper.
2 · 35 min
Ch 3Recipes 4, 5, 6: letter det, |cA|, adjoint.
3 · 25 min
Ch 1Recipes 4, 5, 6: inverse, find C, letters.
4 · tomorrow am
Mock testOne full paper with the timer. Mark it honestly.
How each recipe works
01 · Recipe
How to apply
Numbered steps, plus a "Senang cerita" line and a calculator box.
02 · Drag & watch
See why
Drag the control points. Play the worked example one line at a time.
03 · Do & prove
Do it yourself
Endless questions with hints and full jalan kerja, then 3 quick checks.
One-page formula sheet
2×2 inverse (tukar, tolak, bahagi): a b;c d⁻¹ = 1/ad − bcd -b;-c a
Order flips: (AB)ᵀ = BᵀAᵀ · (AB)⁻¹ = B⁻¹A⁻¹ · Peel: CAB = M → C = MB⁻¹A⁻¹
Elementary matrix = one row move on I · E⁻¹ = the undo move on I
Last row (coef)z = (rhs): coef ≠ 0 unique · 0 = 0 infinite · 0 = non-zero none · homogeneous: always consistent
[A | I] → [I | A⁻¹] · Cofactor Cij = (−1)i+jMij, signs + − + / − + − / + − +
Det rules: swap → × (−1) · factor k out → k × · add k × row → no change · triangular → diagonal product
Properties: |AB| = |A||B| · |Aᵀ| = |A| · |A⁻¹| = 1/|A| · |cA| = cⁿ|A|
Adjoint: A⁻¹ = adj(A)/|A|, adj(A) = C(A)ᵀ · Cramer: x = |Ax|/|A| (replace the x-column by b)