Linear Algebra

Linear Algebra

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Matrix & Rank: A matrix of order m×n has rank ρ(A) = number of non-zero rows in its Row Echelon Form (REF). Rank-Nullity theorem: ρ(A) + nullity(A) = n, where nullity = dimension of null space.

Eigenvalues: For matrix A, solve the characteristic equation: |A − λI| = 0. For a 2×2 matrix: λ² − tr(A)λ + det(A) = 0, where tr(A) = a₁₁ + a₂₂. The sum of eigenvalues = tr(A), and product of eigenvalues = det(A).

Key Formulas:

• det(AB) = det(A)·det(B)
• (AB)ᵀ = BᵀAᵀ
• A⁻¹ = adj(A)/det(A) when det(A) ≠ 0
• Cayley-Hamilton: A satisfies its own characteristic equation

Consistency: System Ax = B has solutions iff ρ(A) = ρ([A|B]).

GATE Tips: Rank questions appear annually (1–2 marks). Always verify ρ(A) = ρ([A|B]) for consistency. For eigenvalue problems, check trace-sum as a quick verification. Cayley-Hamilton enables inverse and power calculations without long division.

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🟡 Standard — Regular Study (2d–2mo)

Matrix Operations and Rank

A matrix A of order m×n transforms ℝⁿ → ℝᵐ. The rank ρ(A) equals the maximum number of linearly independent row or column vectors. Row reduction to REF is the standard tool: transform [A|B] using elementary row operations (swap rows, multiply by a nonzero scalar, add a multiple of one row to another).

Solving Linear Systems Ax = B

Compute both ρ(A) and ρ([A|B]):

• If ρ(A) ≠ ρ([A|B]): inconsistent — no solution
• If ρ(A) = ρ([A|B]) = n: unique solution
• If ρ(A) = ρ([A|B]) < n: infinite solutions, with n − ρ free parameters

Gaussian elimination (forward elimination + back substitution) is the GATE-preferred method; computing A⁻¹ and multiplying is rarely efficient.

Eigenvalues and Eigenvectors

For eigenvalue λ, solve (A − λI)v = 0. The characteristic polynomial p(λ) = det(A − λI) has degree n with n roots counted with multiplicity. A symmetric n×n matrix always has n real eigenvalues.

Cayley-Hamilton Theorem

Every n×n matrix A satisfies its own characteristic equation: p(A) = 0. This provides a direct path to computing A⁻¹ and Aᵏ without repeated matrix multiplication.

Diagonalization

A is diagonalizable iff A possesses n linearly independent eigenvectors. When true, P⁻¹AP = D where D = diag(λ₁, λ₂, …, λₙ) and P is the eigenvector matrix (eigenvectors as columns). A symmetric matrix is guaranteed diagonalizable via an orthogonal matrix (P⁻¹ = Pᵀ).

Linear Independence and Vector Spaces

A set {v₁, v₂, …, vₖ} is linearly independent if c₁v₁ + c₂v₂ + ⋯ + cₖvₖ = 0 implies all cᵢ = 0. The dimension of a vector space equals the cardinality of any basis. For subspace W ⊂ V, dim(W) ≤ dim(V) with equality only when W = V.

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🔴 Extended — Deep Study (3mo+)

Edge Cases and Common Traps

The characteristic equation is |A − λI| = 0 — subtract λ only from diagonal entries. A nilpotent matrix (Aᵏ = 0 for some k) has all eigenvalues equal to zero and cannot be diagonalized unless it is the zero matrix. The geometric multiplicity (dimension of eigenspace) is always ≤ the algebraic multiplicity (root multiplicity in characteristic polynomial); equality for every eigenvalue is both necessary and sufficient for diagonalizability.

Mechanism: Inverse via Cayley-Hamilton

Given A, find p(λ) = det(A − λI) = (−1)ⁿ(λⁿ + cₙ₋₁λⁿ⁻¹ + ⋯ + c₁λ + c₀). The Cayley-Hamilton equation p(A) = 0 gives Aⁿ + cₙ₋₁Aⁿ⁻¹ + ⋯ + c₁A + c₀I = 0. Multiplying by A⁻¹ (valid when det(A) ≠ 0) isolates A⁻¹ = −(1/c₀)(Aⁿ⁻¹ + cₙ₋₁Aⁿ⁻² + ⋯ + c₁I), bypassing direct adjugate computation.

Connections to Adjacent Topics

In differential equations, solving x′ = Ax relies on eigenvalues of A; repeated eigenvalues demand generalized eigenvectors. The minimal polynomial m(λ) (smallest degree monic divisor of p(λ) with m(A) = 0) determines whether diagonalization is possible and appears in canonical forms. In numerical analysis, the power method locates the dominant eigenvalue through repeated matrix-vector multiplication. Linear transformations T:V → W satisfy T(c₁v₁ + c₂v₂) = c₁T(v₁) + c₂T(v₂); the matrix representation depends on the basis chosen for domain and codomain.

Common Mistakes

Subtracting λ from all entries instead of only the diagonal leads to wrong characteristic polynomials. Applying Cayley-Hamilton without first verifying that p(A) = 0 holds for the given matrix yields incorrect results. Inconsistent systems are sometimes incorrectly treated as having infinite solutions when ρ(A) ≠ ρ([A|B]). When computing eigenvectors, the matrix (A − λI) is singular — Gaussian elimination still applies but back-substitution terminates early because at least one row becomes all zeros.

Practice Prompts

1. For A = [[2, 1], [1, 2]], compute eigenvalues using |A − λI| = 0, find eigenvectors for each λ, verify trace-sum and determinant-product, construct P and confirm P⁻¹AP = diag(λ₁, λ₂), then find A¹⁰ using diagonalization without direct multiplication.

2. Classify the system x₁ + 2x₂ = 3, 2x₁ + 4x₂ = 6, 3x₁ + 6x₂ = k for k = 7, k = 9, and k = 10. Determine ρ(A), ρ([A|B]), solution count for each case, and express the general solution as a parametric vector when infinite solutions exist.

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