Advanced Engineering Mathematics - Cheat Sheet
Your last-minute revision companion for complex numbers, matrices, determinants, Cramer's rule, and vector operations. Every formula, definition, and critical concept you need in the final 30 minutes before the exam.
Complex Numbers: Forms and Operations
Formulas
z = a + bi
Meaning
Rectangular form: a = real part, b = imaginary part, i = √(−1)
Watch out
Students forget i² = −1; always simplify imaginary products immediately.
When to use
Starting point for most complex number problems; converting from polar or exponential forms.
|z| = √(a² + b²)
Meaning
Magnitude (modulus) of z; the distance from origin to z in the complex plane.
Watch out
Common error: |z| is NEVER negative; always take the positive square root.
When to use
Always required when converting to polar or exponential form; needed for division/multiplication in polar form.
θ = arctan(b/a) [adjust for quadrant]
Meaning
Argument (angle); θ is measured counterclockwise from the positive real axis in degrees or radians.
Watch out
Quadrant errors: Q1 (a>0, b>0) use arctan directly; Q2 (a<0, b>0) add 180°; Q3 (a<0, b<0) add 180°; Q4 (a>0, b<0) add 360° or use negative angle.
When to use
When converting rectangular to polar form; must check which quadrant a and b place z in.
z = r∠θ = r(cos θ + i sin θ)
Meaning
Polar form: r = |z| magnitude, θ = argument in degrees or radians.
Watch out
Ensure θ is in the correct range (0° to 360° or −180° to 180°); radian vs degree consistency.
When to use
Multiplication, division, powers, and roots are MUCH simpler in polar form.
z = r·e^(iθ)
Meaning
Exponential (Euler) form: e^(iθ) = cos θ + i sin θ.
Watch out
Requires θ in RADIANS; not degree mode. Easy to confuse with polar notation.
When to use
Advanced work with differential equations, signal processing; equivalent to polar form.
z₁ · z₂ = r₁r₂ ∠(θ₁ + θ₂) [polar form]
Meaning
Multiply magnitudes; add arguments.
Watch out
Forget to reduce final angle to 0°–360° range; students mix up 'add angles' with 'add real parts'.
When to use
Always use polar form for multiplication; rectangular form is tedious.
z₁ ÷ z₂ = (r₁/r₂) ∠(θ₁ − θ₂) [polar form]
Meaning
Divide magnitudes; subtract arguments.
Watch out
Subtraction of angles can yield negative angles; adjust to 0°–360° if needed.
When to use
Division is vastly simpler in polar form than rectangular.
z^n = r^n ∠(nθ) [De Moivre's Theorem]
Meaning
Raise magnitude to power n; multiply argument by n.
Watch out
The final angle may exceed 360°; always reduce to standard range. Do NOT expand binomially.
When to use
Powers of complex numbers; infinitely simpler than expanding (a+bi)^n in rectangular form.
z^(1/n) = r^(1/n) ∠[(θ + 360°k)/n] for k = 0, 1, ..., n−1
Meaning
n distinct nth roots; magnitude root is r^(1/n); arguments are (θ + 360k°)/n for each k.
Watch out
There are EXACTLY n distinct roots; students often forget to generate all k values. Missing even one root is a loss of marks.
When to use
Finding square roots, cube roots, etc. of complex numbers.
(a + bi)(a − bi) = a² + b²
Meaning
Product of a complex number and its conjugate equals the square of its magnitude (real number).
Watch out
Conjugate of a+bi is a−bi (flip sign of imaginary part). Easy to forget the sign flip.
When to use
Rationalizing denominators in complex division; always yields a real result.
Common values
Value
0°, 30°, 45°, 60°, 90°, 180°, 270°
Symbol
θ
Quantity
Common argument angles
Value
sin 30° = 0.5, cos 30° = √3/2; sin 45° = √2/2, cos 45° = √2/2; sin 60° = √3/2, cos 60° = 0.5; sin 90° = 1, cos 90° = 0
Symbol
sin θ, cos θ
Quantity
sin and cos at standard angles
Value
1 radian ≈ 57.3°; 1° ≈ 0.0175 rad
Symbol
rad ↔ °
Quantity
Conversion factor
Important facts
- i = √(−1); i² = −1; i³ = −i; i⁴ = 1 (cycle repeats every 4 powers)
- For any complex number z, z · (conjugate of z) = |z|² (always real and positive)
- In polar form, multiplication is easiest: just multiply magnitudes and add angles
- De Moivre's theorem: (r∠θ)^n = r^n∠(nθ); works for any real or integer exponent
- The n nth roots of a complex number are equally spaced around a circle; angles differ by 360°/n
- Conversion: |z| = √(a² + b²); θ = arctan(b/a) with quadrant adjustment
- Euler's formula: e^(iθ) = cos θ + i sin θ; fundamental link between exponential and trigonometric forms
Key definitions
Complex number
Example
3 + 4i (real part 3, imaginary part 4)
Number of the form a + bi where a, b are real and i² = −1.
Magnitude / Modulus
Example
|3 + 4i| = √(9 + 16) = 5
Distance from origin to point z in complex plane; |z| = √(a² + b²).
Argument
Example
arg(3 + 4i) ≈ 53.13°
Angle from positive real axis to point z, measured counterclockwise; typically 0° to 360° or −180° to 180°.
Conjugate
Example
Conjugate of 2 + 3i is 2 − 3i
Complex number with imaginary part negated; conjugate of a+bi is a−bi.
Polar form
Example
3 + 4i = 5∠53.13°
Representation as r∠θ or r(cos θ + i sin θ); used for multiplication, division, powers.
Diagrams to know
- Argand diagram (complex plane) with real axis horizontal and imaginary axis vertical
- Vector representation of complex number from origin to point (a, b)
- Geometric interpretation of multiplication (spiral rotation by angle θ₂) and division (spiral contraction)
Matrices and Determinants
Formulas
det(2×2) = |a b| = ad − bc
|c d|
Meaning
For a 2×2 matrix, determinant = (product of main diagonal) − (product of anti-diagonal).
Watch out
Order matters: ad − bc, NOT ad + bc. Sign errors are the #1 mistake.
When to use
Calculating determinant for 2×2 systems; foundation for larger matrices and Cramer's rule.
det(3×3) = a(ei−fh) − b(di−fg) + c(dh−eg) [cofactor expansion along first row]
Meaning
For 3×3, expand along any row or column using cofactors (−1)^(i+j) times the minor.
Watch out
Sign pattern alternates: +, −, +, − along expansions. Easy to flip a sign; always double-check cofactor signs.
When to use
Solving 3×3 systems with Cramer's rule; finding matrix inverses.
A^(−1) = (1/det A) × adj(A)
Meaning
For invertible square matrix A, inverse = (reciprocal of determinant) × (adjugate matrix).
Watch out
If det A = 0, matrix is SINGULAR; no inverse exists. Adjugate is transpose of cofactor matrix.
When to use
Solving Ax = b as x = A^(−1)b; required when det A ≠ 0.
A^(−1) = (1/(ad−bc)) × |d −b| [2×2 specific]
|−c a|
Meaning
For 2×2: swap a↔d on diagonal, negate b and c off-diagonal, divide by determinant.
Watch out
Must verify det(A) ≠ 0 FIRST. The swap-and-negate pattern is easy to mess up.
When to use
Quick inversion of 2×2 matrices; much faster than cofactor method for small systems.
A · B ≠ B · A [matrix multiplication is NOT commutative]
Meaning
Order of multiplication matters; result depends on which matrix is on the left.
Watch out
Treating matrices like scalars and assuming AB = BA is a critical error in exams.
When to use
Any matrix algebra problem; reminds you to keep track of order.
(m × n) · (n × p) = (m × p)
Meaning
Inner dimensions must match; result has outer dimensions.
Watch out
If inner dimensions don't match, multiplication is UNDEFINED. Always verify dimensions first.
When to use
Checking if two matrices can be multiplied; determining size of product.
det(A·B) = det(A) · det(B)
Meaning
Determinant of a product equals the product of determinants.
Watch out
This is a theorem; NOT true for det(A+B). Do NOT confuse with additive property.
When to use
Simplifying determinant calculations; checking invertibility of products.
det(A^T) = det(A) [transpose property]
Meaning
Determinant of a transposed matrix equals the original determinant.
Watch out
Transpose does NOT change the determinant value; keep this in mind for row/column operations.
When to use
Simplifying problems where you encounter A^T; often used in proofs.
Common values
Value
[1 0]
[0 1]
Symbol
I₂
Quantity
Identity matrix (2×2)
Value
[1 0 0]
[0 1 0]
[0 0 1]
Symbol
I₃
Quantity
Identity matrix (3×3)
Important facts
- Identity matrix I has 1s on the diagonal, 0s elsewhere; A · I = I · A = A for any compatible A
- For a 2×2 matrix, swapping two rows or columns negates the determinant
- Adding a multiple of one row to another does NOT change the determinant
- Multiplying an entire row by k multiplies the determinant by k
- For n×n matrix, det(kA) = k^n · det(A), NOT k · det(A)
- If a matrix has two identical or proportional rows, det = 0 (singular)
- The adjugate matrix is the TRANSPOSE of the cofactor matrix, not the cofactor matrix itself
Key definitions
Matrix
Example
[1 2 3]
[4 5 6] is a 2×3 matrix
Rectangular array of numbers arranged in rows and columns; denoted m × n (m rows, n columns).
Determinant
Example
det([2 3]
[1 4]) = 2(4) − 3(1) = 5
Single scalar value computed from a square matrix; indicates invertibility and scaling effect.
Inverse matrix
Example
For A = [1 2], A^(−1) = [4 −2]
[3 4] [−3 1] (after dividing by det A = −2)
Matrix A^(−1) such that A · A^(−1) = A^(−1) · A = I (identity); exists only if det A ≠ 0.
Minor
Example
For 3×3 matrix, minor M₁₂ is the determinant after deleting row 1 and column 2
Determinant of the matrix formed by deleting one row and one column.
Cofactor
Example
C₁₁ = (−1)^(1+1) M₁₁ = +M₁₁; C₁₂ = (−1)^(1+2) M₁₂ = −M₁₂
Minor with a sign: C_ij = (−1)^(i+j) · M_ij.
Adjugate matrix
Example
If cofactor matrix is C, then adj(A) = C^T
Transpose of the cofactor matrix; denoted adj(A).
Singular matrix
Example
Any matrix with a row of all zeros or two identical rows is singular
Square matrix with det = 0; not invertible; rows or columns are linearly dependent.
Diagrams to know
- Cofactor expansion diagram showing which element pairs to multiply and which signs to apply
- 2×2 determinant calculation diagram (main diagonal minus anti-diagonal)
Cramer's Rule: Solving Linear Systems
Formulas
For Ax = b, x_i = (det A_i) / (det A)
Meaning
Each unknown x_i equals the ratio of two determinants: numerator is A with ith column replaced by b, denominator is det(A).
Watch out
ONLY works if det A ≠ 0. If det A = 0, no unique solution (either no solution or infinite solutions).
When to use
Solving 2×2 or 3×3 linear systems with unique solution; requires det A ≠ 0.
x = (det A₁) / (det A), y = (det A₂) / (det A), z = (det A₃) / (det A) [for 3×3]
Meaning
For system of 3 equations in 3 unknowns: A₁ replaces column 1 of A with b, A₂ replaces column 2, A₃ replaces column 3.
Watch out
Easy to confuse which column gets replaced; always double-check that you're replacing the correct column for each variable.
When to use
Standard application for 3×3 systems; each unknown requires one determinant calculation.
det A ≠ 0 ⟹ unique solution
Meaning
Non-zero determinant of coefficient matrix guarantees exactly one solution.
Watch out
If det A = 0, Cramer's rule fails; the system is either inconsistent or dependent (infinitely many solutions).
When to use
Before applying Cramer's rule; verify the system is solvable with a unique answer.
Important facts
- Cramer's rule applies ONLY to square systems (n equations, n unknowns) where det A ≠ 0
- For a 2×2 system, you need to compute exactly THREE 2×2 determinants (original A, A₁, A₂)
- For a 3×3 system, you need to compute exactly FOUR 3×3 determinants (original A, A₁, A₂, A₃)
- If det A = 0, the system is either inconsistent (no solution) or dependent (infinite solutions); Cramer's rule cannot be used
- The order of replacing columns matters: for x_i, always replace column i (not any other column)
- Cramer's rule is theoretically elegant but computationally expensive for large systems (prefer Gaussian elimination)
Key definitions
Coefficient matrix A
Example
For 2x + 3y = 8 and x + 4y = 9, A = [2 3]
[1 4]
Square matrix of all coefficients from the left side of the system; does not include constants b.
Constant vector b
Example
For the system above, b = [8]
[9]
Column vector of all right-hand-side constants from the equations.
Augmented matrix A_i
Example
For x calculation in 2×2, A₁ = [8 3]
[9 4] (column 1 replaced with b)
Matrix formed by replacing the ith column of A with vector b.
Unique solution
Example
System 2x + 3y = 8, x + 4y = 9 has unique solution x = 1, y = 2
Exactly one set of values satisfying all equations simultaneously; occurs when det A ≠ 0.
Diagrams to know
- 2×2 system setup showing original matrix A and augmented matrices A₁, A₂
- 3×3 system setup showing all four determinant calculations
Vector Operations and Applications
Formulas
a = (a_x, a_y, a_z) [component form]
Meaning
Vector represented as ordered triple of components along x, y, z axes.
Watch out
Do NOT confuse vector (a_x, a_y, a_z) with point (a_x, a_y, a_z); vectors have magnitude AND direction.
When to use
Starting point for all vector calculations; easy to work with componentwise.
|a| = √(a_x² + a_y² + a_z²)
Meaning
Magnitude (length) of vector; distance from origin if a is position vector.
Watch out
Magnitude is ALWAYS non-negative. Magnitude equals zero only if vector is the zero vector (0, 0, 0).
When to use
Whenever you need the 'length' or 'size' of a vector; required for unit vectors.
a · b = a_x·b_x + a_y·b_y + a_z·b_z
Meaning
Dot product (scalar product): sum of products of corresponding components.
Watch out
Dot product yields a SCALAR (single number), never a vector. If result is 0, vectors are perpendicular.
When to use
Computing angles between vectors, checking perpendicularity, projections.
a · b = |a| |b| cos θ
Meaning
Geometric interpretation: dot product equals product of magnitudes times cosine of angle between them.
Watch out
θ is always between 0° and 180° (or 0 and π rad). If dot product is 0, θ = 90° (perpendicular).
When to use
Finding angle θ between two vectors: cos θ = (a · b) / (|a| |b|), then θ = arccos(...).
a × b = |i j k |
|a_x a_y a_z |
|b_x b_y b_z |
Meaning
Cross product (vector product): determinant expansion yielding a vector perpendicular to both a and b.
Watch out
Cross product is a VECTOR (three components), NOT a scalar. a × b ≠ b × a; order matters (reverses direction).
When to use
Finding perpendicular vectors, calculating areas of parallelograms, torque calculations.
a × b = (a_y·b_z − a_z·b_y, a_z·b_x − a_x·b_z, a_x·b_y − a_y·b_x)
Meaning
Component form of cross product after determinant expansion.
Watch out
Easy to mix up signs and component order. Double-check the cyclic pattern: (y·z−z·y, z·x−x·z, x·y−y·x).
When to use
Practical calculation of cross product; often faster than full determinant notation.
|a × b| = |a| |b| sin θ
Meaning
Magnitude of cross product: product of magnitudes times sine of angle.
Watch out
If a × b = 0, then a and b are PARALLEL (θ = 0° or 180°), not perpendicular.
When to use
Area of parallelogram spanned by a and b equals |a × b|; if sin θ = 0, vectors are parallel.
a × b ⊥ a and a × b ⊥ b [perpendicularity property]
Meaning
Cross product is always perpendicular to both input vectors.
Watch out
This is a defining property of cross product; always true for non-parallel vectors.
When to use
Finding normal vectors to a plane; knowing the resulting vector direction.
Unit vector: u = a / |a|
Meaning
Vector with magnitude 1 in the same direction as a.
Watch out
Cannot divide by zero; |a| must be non-zero. Always check the vector is not the zero vector.
When to use
Normalizing vectors; expressing direction without magnitude information.
Projection of a onto b: proj_b(a) = ((a·b) / |b|²) b
Meaning
Component of a in the direction of b; a vector parallel to b.
Watch out
Projection is a VECTOR, not a scalar. If dot product is negative, projection points opposite to b.
When to use
Decomposing forces, finding components in specific directions.
Common values
Value
i = (1, 0, 0); j = (0, 1, 0); k = (0, 0, 1)
Symbol
i, j, k
Quantity
Standard basis vectors
Value
θ = 90° or π/2 radians
Symbol
θ
Quantity
Angle where cos θ = 0 (perpendicular)
Value
θ = 0° or 180° (0 or π radians)
Symbol
θ
Quantity
Angle where sin θ = 0 (parallel)
Important facts
- Dot product of perpendicular vectors equals zero; dot product of parallel vectors equals ±|a||b|
- Cross product of parallel vectors equals zero vector; cross product magnitude equals area of parallelogram
- Right-hand rule: for a × b, curl fingers from a toward b, thumb points in direction of result
- Cross product is anti-commutative: a × b = −(b × a)
- Dot product is commutative: a · b = b · a
- For unit vectors: |u| = 1; scalar multiplication of unit vector changes magnitude but preserves direction
- Projection of a onto b is zero if vectors are perpendicular; projection equals a if vectors are parallel
Key definitions
Vector
Example
a = (3, 4, 0) has magnitude 5 and direction in the xy-plane
Quantity with both magnitude and direction; represented as ordered list of components or arrow in space.
Dot product (scalar product)
Example
(1, 2, 3) · (4, 5, 6) = 1(4) + 2(5) + 3(6) = 32
Binary operation yielding a scalar; measures 'alignment' of two vectors.
Cross product (vector product)
Example
(1, 0, 0) × (0, 1, 0) = (0, 0, 1) (right-hand rule applies)
Binary operation yielding a vector perpendicular to both inputs; non-commutative.
Magnitude
Example
|(3, 4, 0)| = √(9 + 16 + 0) = 5
Length of a vector; always non-negative; |a| = √(a_x² + a_y² + a_z²).
Unit vector
Example
Unit vector in direction of (3, 4, 0) is (3/5, 4/5, 0)
Vector with magnitude exactly 1; used to represent pure direction.
Angle between vectors
Example
Angle between (1, 0, 0) and (1, 1, 0): cos θ = 1/(1·√2) = 1/√2, so θ = 45°
Angle θ ∈ [0°, 180°] found using cos θ = (a · b) / (|a| |b|).
Perpendicular (orthogonal) vectors
Example
(1, 0, 0) and (0, 1, 0) are perpendicular; their dot product is 0
Two vectors a and b are perpendicular if a · b = 0.
Parallel vectors
Example
(2, 4, 6) and (1, 2, 3) are parallel; (1, 2, 3) = ½(2, 4, 6)
Two vectors a and b are parallel if a × b = 0 (or one is a scalar multiple of the other).
Diagrams to know
- Angle between two vectors diagram with dot product and cosine relationship
- Cross product diagram with right-hand rule orientation and perpendicularity to both inputs
- Projection diagram showing vector a decomposed into parallel and perpendicular components relative to b
Must remember
- Complex number magnitude: |z| = √(a² + b²). Always non-negative. Required for polar form and division.
- De Moivre's Theorem: z^n = r^n∠(nθ). Raise magnitude to power n, multiply argument by n. NEVER expand binomially for powers of complex numbers.
- Matrix determinant (2×2): |a b; c d| = ad − bc. Order is critical: main diagonal MINUS anti-diagonal, not plus.
- Cramer's Rule applies ONLY when det(A) ≠ 0. If det(A) = 0, system has no unique solution. Do NOT attempt Cramer's rule on singular matrices.
- Dot product yields a SCALAR; cross product yields a VECTOR. Dot product = 0 means perpendicular; cross product = 0 means parallel.
- Angle between vectors: cos θ = (a · b) / (|a| |b|). Result is always θ ∈ [0°, 180°].
- Cross product determinant: expand along first row with i, j, k basis vectors. Signs alternate (+, −, +). Easy to swap signs.
- nth roots of a complex number: There are EXACTLY n distinct roots, equally spaced by 360°/n. Generate all k = 0, 1, ..., n−1; missing even one root is a major error.
- Matrix inverse exists ONLY if det(A) ≠ 0. Singular matrices cannot be inverted. For 2×2, swap diagonal elements, negate off-diagonal, divide by determinant.
- Quadrant angles for complex numbers: Q1 (a>0, b>0) arctan(b/a) direct; Q2 (a<0, b>0) add 180°; Q3 (a<0, b<0) add 180°; Q4 (a>0, b<0) add 360° or negative.
Last minute tips
- Always verify QUADRANT for complex number arguments before converting to polar form. Arctan(b/a) alone is NOT sufficient; adjust for correct quadrant.
- Before applying Cramer's Rule, COMPUTE det(A) first and confirm it is non-zero. If det(A) = 0, stop immediately—Cramer's rule does not apply.
- For cross product calculation, use the determinant method with i, j, k and verify the sign pattern (positive for i row, negative for j row, positive for k row). Sign errors are extremely common.
- When finding nth roots of complex numbers, systematically generate ALL n values using k = 0, 1, ..., n−1 in the formula z^(1/n) = r^(1/n)∠[(θ + 360k°)/n]. Omitting even one root costs significant marks.
- Distinguish DOT vs CROSS: If you need a scalar (angle, projection magnitude), use dot product; if you need a vector (normal, area vector, torque), use cross product. Mixing these up inverts the entire answer type.
Comparison tables
Dot Product vs Cross Product
| Property | Dot Product (·) | Cross Product (×) |
|---|---|---|
| Result type | Scalar (single number) | Vector (three components) |
| Geometric meaning | Measure of alignment; |a||b|cos θ | Perpendicular vector; magnitude |a||b|sin θ |
| Formula | a · b = a_x·b_x + a_y·b_y + a_z·b_z | a × b = determinant with i, j, k |
| Commutative? | YES: a · b = b · a | NO: a × b = −(b × a) |
| Zero result means | Perpendicular vectors (θ = 90°) | Parallel vectors (θ = 0° or 180°) |
| When to use | Angles, projections, work, power | Normal vectors, areas, torque, orientation |
Complex Number Forms: Conversion and Use
| Form | Notation | Key Variables | Best for |
|---|---|---|---|
| Rectangular | z = a + bi | a (real), b (imaginary) | Addition, subtraction, conjugates |
| Polar | z = r∠θ | r (magnitude), θ (argument) | Multiplication, division, powers, roots |
| Exponential | z = r·e^(iθ) | r, θ (in radians) | Differential equations, advanced applications |
Matrix Determinant Properties and Implications
| Determinant Condition | Matrix Property | Solution to Ax = b |
|---|---|---|
| det(A) ≠ 0 | Non-singular (invertible) | Unique solution: x = A^(−1)b |
| det(A) = 0 | Singular (not invertible) | No solution or infinite solutions (inconsistent or dependent) |
When to Apply Cramer's Rule vs Other Methods
| Condition | Use Cramer's Rule? | Reason |
|---|---|---|
| det(A) ≠ 0, small system (2×2 or 3×3) | YES | Efficient and clean; each unknown from one ratio of determinants |
| det(A) = 0 | NO | Cramer's rule fails; use Gaussian elimination to check consistency |
| Large system (4×4 or bigger) | NO | Computationally expensive; Gaussian elimination is faster |
Diagrams
Advanced Engineering Mathematics: Complete Formula Map
Complete mindmap showing all major formula categories and relationships in Advanced Engineering Mathematics. Use this to identify which section contains the formula you need.
Complex Number Problem-Solving Decision Tree
Flowchart guiding you through which form and operation to use for different complex number problems. Follows the decision path from problem type to solution method.
Cramer's Rule Application Flowchart
Step-by-step decision tree for applying Cramer's rule. Ensures you check all prerequisites (square matrix, non-zero determinant) before attempting the calculation.
Dot Product vs Cross Product: When to Use
Decision flowchart to determine whether dot or cross product is needed for your problem. Prevents confusion between scalar and vector results.