GATE/Engineering Mathematics/Linear Algebra — Matrices, Rank & Eigenvalues
Sign in to track
Hard20 min readEngineering Mathematics

Linear Algebra — Matrices, Rank & Eigenvalues

Linear algebra covers matrix operations, systems of equations, rank, determinants, and eigenvalues. GATE tests rank-nullity, consistency of systems, and eigenvalue properties.

Key Points

  • ·Matrix rank: maximum number of linearly independent rows; rank(A) ≤ min(m,n)
  • ·Rank-nullity theorem: rank(A) + nullity(A) = n (number of columns)
  • ·System Ax = b: consistent iff rank(A) = rank([A|b]); unique iff rank = n
  • ·Eigenvalue λ: Av = λv; find via det(A − λI) = 0
  • ·Sum of eigenvalues = trace(A); product of eigenvalues = det(A)
  • ·Symmetric matrix: all eigenvalues real, eigenvectors orthogonal

What is a Matrix? What is Rank?

Think of a matrix as a set of equations written compactly:

2x + 3y = 7      ┌2 3┐ ┌x┐   ┌7┐
x  + 4y = 5  →  │1 4│ │y│ = │5│   →  Ax = b
                 └   ┘ └ ┘   └ ┘

Rank of a matrix = how many of its equations are truly independent (not redundant).

Imagine three friends all saying "I agree with what she said." That adds no new information — rank is lower than 3.

Finding Rank via Row Reduction

Convert matrix to Row Echelon Form (REF) using Gaussian elimination. Count non-zero rows.

A = ┌1 2 3┐
    │2 4 6│   ← Row 2 = 2 × Row 1 (dependent!)
    │1 0 1┘

Row reduce: R2 ← R2 − 2R1
          ┌1 2 3┐
          │0 0 0│   ← became all zeros
          │1 0 1┘

R3 ← R3 − R1
          ┌1 2 3┐
          │0 0 0│
          │0 −2 −2┘

Two non-zero rows → rank(A) = 2

Solving Systems of Equations: Ax = b

The augmented matrix [A|b] adds the right-hand side as an extra column.

Three possible outcomes:

Case 1: rank(A) < rank([A|b])
→ The equations are CONTRADICTORY → NO SOLUTION
Example: x + y = 2,  x + y = 5  (impossible!)

Case 2: rank(A) = rank([A|b]) = n  (n = number of unknowns)
→ Exactly ONE solution

Case 3: rank(A) = rank([A|b]) < n
→ INFINITELY MANY solutions (n − rank free variables)

Homogeneous system Ax = 0: always has the trivial solution x = 0. Non-trivial solution (x ≠ 0) exists only if det(A) = 0.

Worked Example

Solve: x + y + z = 6, 2x + 2y + 2z = 12, x + 2y + z = 8

Augmented:
┌1 1 1 | 6 ┐
│2 2 2 | 12│   ← R2 = 2×R1 → subtract 2R1
└1 2 1 | 8 ┘   ← R3 → subtract R1

After reduction:
┌1 1 1 | 6┐
│0 0 0 | 0│
└0 1 0 | 2┘

rank(A) = 2, rank([A|b]) = 2, n = 3
→ Infinitely many solutions (3 − 2 = 1 free variable)

From R3: y = 2. From R1: x + 2 + z = 6 → x + z = 4 → x = 4 − z (z is free)

Determinants

The determinant measures "how much the matrix scales space." If det = 0, the matrix squashes space into a lower dimension (rank < n).

2×2 formula:

det ┌a b┐ = ad − bc
    └c d┘

3×3 cofactor expansion along row 1:

det ┌a b c┐ = a·det┌e f┐ − b·det┌d f┐ + c·det┌d e┐
    │d e f│        └h i┘        └g i┘        └g h┘
    └g h i┘

Key properties (GATE frequently tests these):

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)
Swap two rows → sign of det changes

Eigenvalues and Eigenvectors

An eigenvector v of matrix A is a special direction that A does not rotate — it only scales by factor λ (the eigenvalue):

A · v = λ · v

Imagine stretching a rubber sheet. Most points move in complicated ways. But some directions just get stretched or compressed — those are eigenvectors.

How to Find Eigenvalues

Rearrange Av = λv to (A − λI)v = 0. For non-trivial solution:

det(A − λI) = 0   ← "characteristic equation"

Example: Find eigenvalues of A = ┌4 1┐ └2 3┘

A − λI = ┌4−λ  1 ┐
         └ 2  3−λ┘

det = (4−λ)(3−λ) − (1)(2)
    = λ² − 7λ + 12 − 2
    = λ² − 7λ + 10
    = (λ−5)(λ−2) = 0

Eigenvalues: λ₁ = 5, λ₂ = 2

Verify using shortcuts: - Trace = 4 + 3 = 7 = λ₁ + λ₂ = 5 + 2 ✓ - Det = 4×3 − 1×2 = 10 = λ₁ × λ₂ = 5 × 2 ✓

Important Eigenvalue Properties

If A has eigenvalue λ Then...
Aᵏ has eigenvalue λᵏ
A⁻¹ has eigenvalue 1/λ
A + cI has eigenvalue λ + c
A is triangular diagonal entries are eigenvalues
A is symmetric all eigenvalues are real
A is idempotent (A²=A) eigenvalues ∈ {0, 1}

Quick Check

Q1. Matrix A has rank 2, and has 5 columns. What is the nullity?

Answer: nullity = n − rank = 5 − 2 = 3.

Q2. Eigenvalues of a 3×3 matrix are 1, 2, 3. What is the trace?

Answer: trace = sum of eigenvalues = 1 + 2 + 3 = 6.

Q3. det(3A) where A is a 2×2 matrix with det(A) = 4?

Answer: det(kA) = k² det(A) = 9 × 4 = 36 for a 2×2 matrix.

Key Formulas

  • Rank-Nullity: rank(A) + nullity(A) = n (number of columns)
  • Characteristic equation: det(A − λI) = 0
  • Trace = sum of eigenvalues: tr(A) = λ₁ + λ₂ + ... + λₙ
  • Det = product of eigenvalues: det(A) = λ₁ × λ₂ × ... × λₙ

GATE Exam Tips

  • ★Use trace and det shortcuts to verify eigenvalues — faster than solving the full characteristic polynomial
  • ★Rank of a matrix = non-zero rows after row reduction — practice Gaussian elimination
  • ★Consistent system: rank(A) = rank([A|b]). Unique solution also needs rank = number of unknowns
  • ★det(kA) = kⁿ det(A) for n×n matrix — the n exponent is commonly forgotten

Finished reading this topic?

Mark it complete to track your study progress.

Sign in to track