Linear Algebra
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Linear Algebra — Key Facts for GATE Engineering Mathematics
Core Topics:
| Topic | Key Concepts |
|---|---|
| Matrix Operations | Addition, multiplication, transpose, trace |
| Determinants | Properties, evaluation, adjoint, inverse |
| Rank of Matrix | Row rank = Column rank, rank-nullity theorem |
| System of Equations | Consistent, inconsistent, unique/infinite solutions |
| Eigenvalues & Eigenvectors | Characteristic equation, diagonalization |
| Vector Spaces | Basis, dimension, linear independence |
Quick Formulas:
- det(AB) = det(A) × det(B)
- rank(A) + nullity(A) = n (for n×n matrix)
- Eigenvalues of A⁻¹ = 1/eigenvalues of A
- Cayley-Hamilton Theorem: A satisfies its own characteristic equation
⚡ GATE Tip: In GATE, Linear Algebra questions are worth 8-12 marks out of 100. Focus on eigenvalues, rank, and solving linear systems.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Linear Algebra — Detailed Study Guide
Matrices — Types and Properties
Types of Matrices
| Type | Definition | Example |
|---|---|---|
| Row Matrix | 1×n | [1 2 3] |
| Column Matrix | m×1 | [1; 2; 3] |
| Square Matrix | m×m | [1 2; 3 4] |
| Zero Matrix | All elements 0 | [0 0; 0 0] |
| Identity Matrix | 1’s on diagonal, 0 elsewhere | Iₙ |
| Diagonal Matrix | Non-diagonal = 0 | diag(1,2,3) |
| Symmetric Matrix | A = Aᵀ | [1 2; 2 3] |
| Skew-Symmetric | A = -Aᵀ | [0 2; -2 0] |
| Orthogonal Matrix | AᵀA = I | Rotations |
| Singular Matrix | det(A) = 0 | — |
| Non-singular | det(A) ≠ 0 | — |
| Hermitian | A = Āᵀ (conjugate transpose) | — |
| Unitary | ĀᵀA = I | — |
Matrix Operations
Addition: A + B (same dimensions) Scalar Multiplication: kA Multiplication: (i,j) element = row i of A · column j of B Transpose: Aᵀ (rows become columns) Trace: Tr(A) = sum of diagonal elements Conjugate: Ā (each element conjugated)
Key Properties:
| Property | Formula |
|---|---|
| (Aᵀ)ᵀ = A | Transpose twice |
| (A + B)ᵀ = Aᵀ + Bᵀ | Additivity |
| (AB)ᵀ = BᵀAᵀ | Reversal rule |
| Tr(AB) = Tr(BA) | Cyclic property |
⚡ GATE Memory: (AB)ᵀ = BᵀAᵀ — the order reverses!
Determinants
Definition
For 2×2: det(A) = |a b; c d| = ad - bc
For 3×3 (Sarrus’ Rule):
det = a(ei - fh) - b(di - fg) + c(dh - eg)
= aei + bfg + cdh - ceg - bdi - afhProperties of Determinants
| Property | Effect |
|---|---|
| Row/column swap | Changes sign |
| Two equal rows | det = 0 |
| Row of zeros | det = 0 |
| Multiply row by k | det multiplied by k |
| Add multiple of row to another | det unchanged |
| det(Aᵀ) = det(A) | Invariant under transpose |
| det(AB) = det(A)·det(B) | Multiplicative |
| det(A⁻¹) = 1/det(A) | If A is invertible |
Adjoint and Inverse
Adjugate Matrix: transpose of cofactor matrix
- adj(A) = Cof(A)ᵀ
Inverse: $$A^{-1} = \frac{1}{\det(A)} \cdot adj(A)$$
A matrix is invertible ⟺ det(A) ≠ 0 ⟺ rank(A) = n
⚡ GATE PYQ: “If A is a 3×3 matrix with det(A) = 5, find det(2A).” Answer: det(2A) = 2³ × det(A) = 8 × 5 = 40
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Linear Algebra — Complete Notes for GATE
Rank of a Matrix
Definition
The rank of a matrix is the maximum number of linearly independent rows (or columns).
Methods to Find Rank
Method 1: Echelon Form
- Perform elementary row operations to get row echelon form
- Count non-zero rows = rank
Method 2: Minor Method
- Find largest square submatrix with non-zero determinant
- Size of that submatrix = rank
Elementary Row Operations (don’t change rank):
- Swap two rows
- Multiply a row by non-zero scalar
- Add multiple of one row to another
Rank-Nullity Theorem
For an m×n matrix A: $$\text{rank}(A) + \text{nullity}(A) = n$$
where nullity = dimension of null space (kernel)
System of Linear Equations
Matrix Form: Ax = b
Solutions:
| Condition | Type of Solution |
|---|---|
| rank(A) = rank([A|b]) = n | Unique solution |
| rank(A) = rank([A|b]) < n | Infinite solutions |
| rank(A) ≠ rank([A|b]) | No solution (inconsistent) |
Augmented Matrix: [A|b] — add b as last column
⚡ GATE Worked Example: Solve: x + y + z = 6 2x + 3y + z = 10 x + 2y + 3z = 14
Matrix form: A = [1 1 1; 2 3 1; 1 2 3], b = [6; 10; 14] det(A) = 1(9-2) - 1(6-1) + 1(4-3) = 7 - 5 + 1 = 3 ≠ 0 Unique solution exists.
Using Cramer’s rule or Gaussian elimination: x = 1, y = 2, z = 3
Eigenvalues and Eigenvectors
Definition
For a square matrix A (n×n), a scalar λ is an eigenvalue if: $$A\mathbf{x} = \lambda \mathbf{x}$$
where x ≠ 0 is the eigenvector corresponding to λ.
Characteristic Equation
$$\det(A - \lambda I) = 0$$
This gives a polynomial of degree n — the characteristic polynomial. Roots = eigenvalues.
Properties
| Property | Formula/Rule |
|---|---|
| Sum of eigenvalues | tr(A) |
| Product of eigenvalues | det(A) |
| Eigenvalues of A⁻¹ | 1/λ₁, 1/λ₂, … (if A invertible) |
| Eigenvalues of Aᵀ | Same as eigenvalues of A |
| Eigenvalues of A² | λ₁², λ₂², … |
| Cayley-Hamilton | A satisfies det(A - λI) = 0 |
Cayley-Hamilton Theorem
Every square matrix satisfies its own characteristic equation.
Example: If char poly is λ³ - 5λ² + 2λ + 7 = 0 Then A³ - 5A² + 2A + 7I = 0
Diagonalization
A matrix A is diagonalizable if A = PDP⁻¹ where D is diagonal.
Condition: A has n linearly independent eigenvectors.
P: Matrix whose columns are eigenvectors. D: Diagonal matrix with eigenvalues.
⚡ GATE PYQ: For A = [4 1; 2 3], find eigenvalues. Solution: det(A - λI) = (4-λ)(3-λ) - 2 = λ² - 7λ + 10 = (λ-5)(λ-2) = 0 Eigenvalues: λ₁ = 5, λ₂ = 2
Vector Spaces
Definitions
Vector Space V over F:
- V is closed under addition and scalar multiplication
- 8 axioms satisfied (associativity, commutativity, identity, inverse, etc.)
Subspace: Subset of V that is itself a vector space.
Span: All linear combinations of a set of vectors.
Linear Independence: Vectors v₁, v₂, …, vₙ are linearly independent if: c₁v₁ + c₂v₂ + … + cₙvₙ = 0 implies c₁ = c₂ = … = cₙ = 0
Basis and Dimension
Basis: A set of linearly independent vectors that span V. Dimension: Number of vectors in any basis (same for all bases).
Standard Bases:
- Rⁿ: Standard basis e₁, e₂, …, eₙ
- Pₙ: {1, x, x², …, xⁿ} has dimension n+1
Inner Product Spaces
Inner Product on V:
- ⟨u, v⟩: V × V → F
- Properties: Positive definite, linearity, conjugate symmetry
Norm: ‖v‖ = √⟨v, v⟩ Orthogonality: ⟨u, v⟩ = 0
Gram-Schmidt Orthogonalization: Given {v₁, v₂, …, vₙ}, orthogonalize:
- u₁ = v₁
- u₂ = v₂ - proj_{u₁}(v₂)
- u₃ = v₃ - proj_{u₁}(v₃) - proj_{u₂}(v₃)
- etc.
Then normalize to get orthonormal basis.
GATE-Style Practice Questions
1. The rank of matrix [1 2 3; 2 4 6; 3 6 9] is:
(a) 0 (b) 1 (c) 2 (d) 3
Answer: (b) 1
Solution: Row 2 = 2×Row 1, Row 3 = 3×Row 1 → only 1 linearly independent row
2. If eigenvalues of A are 1, 2, 3, then eigenvalues of A² are:
(a) 1, 2, 3 (b) 1, 4, 9 (c) 1, 1, 1 (d) 2, 3, 4
Answer: (b) 1, 4, 9
Solution: If λ is eigenvalue of A, then λ² is eigenvalue of A²
3. For which value of k does the system have infinite solutions?
x + y + z = 1
2x + 2y + 2z = k
(a) k = 1 (b) k = 2 (c) k = 3 (d) k = 0
Answer: (b) k = 2
Solution: Second equation = 2×(first equation) only if k = 2
4. The trace of matrix [3 1; -1 2] is:
(a) 3 (b) 2 (c) 5 (d) 1
Answer: (c) 5
Solution: Trace = sum of diagonal = 3 + 2 = 5
5. If A is singular, then:
(a) det(A) > 0 (b) det(A) < 0 (c) det(A) = 0 (d) det(A) ≠ 0
Answer: (c) det(A) = 0
Solution: Singular means not invertible → determinant = 0⚡ GATE Strategy: For Linear Algebra in GATE, expect 2-3 questions from eigenvalues, rank, and systems of equations. Always check if matrix is singular/non-singular first.
Content adapted based on your selected roadmap duration. Switch tiers using the selector above.