1.1 Matrices — Key Definitions
| Term | Definition / Property |
|---|---|
| Rank of a matrix | Number of linearly independent rows (or columns); = number of non-zero rows in row echelon form |
| Singular matrix | det(A) = 0; rank < n; system Ax = b may have no or infinite solutions |
| Symmetric | A = Aᵀ |
| Skew-symmetric | A = −Aᵀ; diagonal elements = 0 |
| Orthogonal | AAᵀ = I; Aᵀ = A⁻¹; columns are orthonormal |
| Idempotent | A² = A |
| Nilpotent | Aᵏ = 0 for some integer k |
1.2 Determinants
det(AB) = det(A) · det(B)
det(Aᵀ) = det(A)
det(kA) = kⁿ · det(A) for n×n matrix
det(A⁻¹) = 1 / det(A)
If two rows are identical or proportional → det = 0
Row/column swap → sign of det changes
det(Aᵀ) = det(A)
det(kA) = kⁿ · det(A) for n×n matrix
det(A⁻¹) = 1 / det(A)
If two rows are identical or proportional → det = 0
Row/column swap → sign of det changes
1.3 System of Linear Equations
Ax = b; augmented matrix [A|b]
ρ(A) = rank of A; ρ([A|b]) = rank of augmented matrix
Consistency (Rouché–Capelli theorem):
ρ(A) ≠ ρ([A|b]) → No solution (inconsistent)
ρ(A) = ρ([A|b]) = n → Unique solution
ρ(A) = ρ([A|b]) < n → Infinitely many solutions (n − ρ free variables)
ρ(A) = rank of A; ρ([A|b]) = rank of augmented matrix
Consistency (Rouché–Capelli theorem):
ρ(A) ≠ ρ([A|b]) → No solution (inconsistent)
ρ(A) = ρ([A|b]) = n → Unique solution
ρ(A) = ρ([A|b]) < n → Infinitely many solutions (n − ρ free variables)
1.4 Eigenvalues and Eigenvectors
Ax = λx; characteristic equation: det(A − λI) = 0
Properties of eigenvalues:
Sum of eigenvalues = trace(A) = Σa_ii
Product of eigenvalues = det(A)
Eigenvalues of Aᵀ = eigenvalues of A
Eigenvalues of A⁻¹ = 1/λ (if A invertible)
Eigenvalues of Aᵏ = λᵏ
Symmetric matrix → all eigenvalues are real
Skew-symmetric → eigenvalues are zero or purely imaginary
Orthogonal matrix → |λ| = 1
Properties of eigenvalues:
Sum of eigenvalues = trace(A) = Σa_ii
Product of eigenvalues = det(A)
Eigenvalues of Aᵀ = eigenvalues of A
Eigenvalues of A⁻¹ = 1/λ (if A invertible)
Eigenvalues of Aᵏ = λᵏ
Symmetric matrix → all eigenvalues are real
Skew-symmetric → eigenvalues are zero or purely imaginary
Orthogonal matrix → |λ| = 1
📝 ESE Tip: Rank-nullity theorem: rank(A) + nullity(A) = n (number of columns). Cayley–Hamilton theorem: every matrix satisfies its own characteristic equation — A² = (trace)A − (det)I for 2×2.