1. Matrices & Determinants
Operations: add, multiply, transpose, inverse
Rank: max independent rows/columns
Types: singular (det=0), orthogonal (AᵀA=I), symmetric
Eigenvalues: det(A − λI) = 0
Eigenvectors: (A − λI)x = 0
Cayley-Hamilton: every matrix satisfies its characteristic equation.
2. Differential Equations
First order: variable separable, linear (IF = e∫P dx), exact
Second order linear: homogeneous (aux. equation), particular integral
Applications: growth/decay, circuits (RLC), vibrations
3. Laplace Transforms
L{f(t)} = ∫₀^∞ e^{−st} f(t) dt
Standard pairs
- 1 → 1/s
- tⁿ → n!/sⁿ⁺¹
- e^{at} → 1/(s−a)
- sin at → a/(s²+a²)
- cos at → s/(s²+a²)
Use: convert DE to algebraic equation, solve, inverse Laplace.