CELE Engineering Mathematics — Advanced Engineering MathematicsRevision Notes

Formulas

Exam Tips

• Memorise the 5 special angles and their exact cos/sin values for quick polar-to-rectangular conversions (0°, 30°, 45°, 60°, 90°).
• When the problem asks for the 'modulus' or 'absolute value', compute |z| = √(a² + b²) immediately.
• For multiplication or division of two complex numbers given in rectangular form, convert to polar first to save time.
• Check: sum of all n-th root angles = (n−1)×180° — a quick self-verification trick.
• Board exams frequently pair De Moivre with binomial expansion questions — recognise the shortcut.

Key Points

• A complex number z = a + bi consists of a real part (a) and an imaginary part (b), where i = √(−1).
• Three equivalent representations: Rectangular (a + bi), Polar (r∠θ), and Exponential (re^(iθ)).
• Magnitude (modulus): |z| = √(a² + b²).
• Argument (angle): θ = tan⁻¹(b/a) — always verify the correct quadrant by checking signs of a and b.
• Multiplication is easiest in polar form: multiply magnitudes, add angles.
• Division in polar form: divide magnitudes, subtract angles.
• De Moivre's Theorem for powers: z^n = r^n ∠ nθ.
• n-th roots: z^(1/n) = r^(1/n) ∠ [(θ + 360°k)/n], for k = 0, 1, …, n−1 — this gives exactly n distinct roots.
• Complex conjugate of z = a + bi is z* = a − bi; product z·z* = |z|² = a² + b².
• Euler's formula: e^(iθ) = cos θ + i sin θ — fundamental link between exponential and polar forms.

Definitions

Section Title

Complex Numbers

Common Mistakes

• Ignoring quadrant when computing θ — always check signs of a and b before applying tan⁻¹.
• Using θ in degrees when formula requires radians (or vice versa) — be consistent throughout a problem.
• Forgetting to include ALL n distinct n-th roots — a cube root always has 3 answers.
• Multiplying complex numbers in rectangular form when polar form is faster and less error-prone.
• Treating i² as +1 instead of −1 when expanding products in rectangular form.

Formulas

Exam Tips

• For 2×2 problems, det = ad − bc is the fastest computation — do it first before anything else.
• When expanding a 3×3 determinant, choose the row or column with the most zeros to minimise arithmetic.
• Double-check matrix dimensions before multiplying — incompatible dimensions is a common error that wastes time.
• Memorise the 2×2 inverse formula exactly; it appears in 3–5 board-exam problems per subject cycle.
• If det A = 0 is found during Cramer's rule, immediately state 'no unique solution exists' and verify if the system is consistent or inconsistent.

Key Points

• A matrix is a rectangular array of numbers; its size is described as m×n (m rows, n columns).
• 2×2 determinant formula: det A = ad − bc for A = [[a, b], [c, d]].
• 3×3 determinant is computed by cofactor expansion along any row or column (choose the row/column with the most zeros).
• Matrix multiplication: (m×n)(n×p) = (m×p); the inner dimensions must match.
• Matrix multiplication is NOT commutative in general: AB ≠ BA.
• Identity matrix I satisfies AI = IA = A for any square matrix A.
• 2×2 inverse: A⁻¹ = (1/det A) × [[d, −b], [−c, a]].
• A matrix is singular (no inverse) when det A = 0.
• Transpose A^T is obtained by swapping rows and columns: (A^T)ᵢⱼ = Aⱼᵢ.
• Row operations (scaling, swapping, adding multiples of rows) are used in Gaussian elimination to solve systems.

Definitions

Section Title

Matrices and Determinants

Common Mistakes

• Reversing the elements when computing the 2×2 inverse — memorise the swap: main diagonal swaps, off-diagonal elements negate.
• Confusing matrix multiplication order: (AB)ᵢⱼ = Σ aᵢₖ bₖⱼ; always row-of-first times column-of-second.
• Forgetting that AB ≠ BA in general for matrices.
• Sign errors in cofactor expansion — the checkerboard sign pattern (+−+/−+−/+−+) must be applied correctly.
• Attempting Cramer's rule when det A = 0 — system has no unique solution.

Formulas

Exam Tips

• Always compute det(A) FIRST — if it equals zero, stop and re-examine the problem.
• When the problem asks only for x (or only for y), compute only det(A) and det(Aₓ) — no need to find all unknowns.
• Write the system in standard form (all unknowns on the left, constants on the right) before forming the matrices.
• Label your matrices clearly (A, A₁, A₂) to avoid substitution errors under time pressure.

Key Points

• Cramer's rule solves a linear system Ax = b by expressing each unknown as a ratio of determinants.
• xᵢ = det(Aᵢ) / det(A), where Aᵢ is obtained by replacing the i-th column of A with the constant vector b.
• Valid ONLY when det(A) ≠ 0 (non-singular system).
• For a 2×2 system: two determinant computations give x and y directly.
• For a 3×3 system: four determinant computations (det A, det A₁, det A₂, det A₃) give x, y, z.
• Cramer's rule is elegant but computationally intensive for large systems — on board exams, it is almost always tested at the 2×2 or 3×3 level.
• Preferred over elimination when the problem explicitly asks for ONE specific unknown — compute only the required determinant ratio.

Definitions

Section Title

Cramer's Rule

Common Mistakes

• Replacing the wrong column when forming Aᵢ — always replace column i (the column corresponding to the unknown xᵢ) with vector b.
• Computing det(A) = 0 and still proceeding with Cramer's rule — the rule is undefined in this case.
• Arithmetic sign errors when computing 3×3 determinants under exam pressure — use the expansion method systematically.
• Mixing up which unknown corresponds to which column when setting up A₁, A₂, A₃.

Formulas

Exam Tips

• When asked to find the angle between two vectors, use the dot product formula — it directly gives cos θ.
• When asked whether two vectors are perpendicular, compute the dot product: if it equals zero, they are perpendicular.
• For cross product computation, write out the 3×3 determinant explicitly to avoid sign errors.
• The area of a parallelogram = |a × b|; the area of a triangle = ½|a × b| — common geometry problem variation.
• Memorise: dot product → scalar, result tells angle; cross product → vector, magnitude tells area.

Key Points

• A vector has both magnitude and direction; in 3D: v = vₓi + vyj + vzk or written as (vₓ, vy, vz).
• Magnitude: |v| = √(vₓ² + vy² + vz²).
• Unit vector: v̂ = v / |v| — always has magnitude 1.
• Dot product (scalar product): a · b = aₓbₓ + ayby + azbz = |a||b|cos θ.
• Dot product is zero if and only if the two vectors are perpendicular (orthogonal).
• Cross product (vector product): a × b = determinant expansion using i, j, k unit vectors.
• Magnitude of cross product: |a × b| = |a||b|sin θ — equals the area of the parallelogram formed by a and b.
• Cross product is zero if and only if the vectors are parallel (or one is the zero vector).
• Cross product is anticommutative: a × b = −(b × a).
• Scalar triple product: a · (b × c) = volume of the parallelepiped formed by the three vectors.

Definitions

Section Title

Vectors

Common Mistakes

• Confusing dot product (scalar result) with cross product (vector result) — always identify what the problem is asking for.
• Computing cross product components with wrong signs — use the determinant method systematically to avoid sign errors.
• Forgetting that a × b ≠ b × a — cross product is anticommutative (b × a = −a × b).
• Using 2D magnitude formula √(a²+b²) for a 3D vector with three components.
• Dividing by zero when computing unit vectors — always check |v| ≠ 0 first.