Rank of Matrix – Engineering Maths for GATE, ESE, AE/JE | Priyanka Ma’am

The rank of a matrix is a fundamental concept in Linear Algebra under Engineering Mathematics. It is widely used in solving systems of equations and is frequently asked in GATE, ESE, AE/JE exams.

Basic Concept

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.

It is denoted as: Rank(A)

Important Points

• Rank ≤ min(number of rows, number of columns)
• If determinant ≠ 0 (for square matrix) → Rank = order of matrix
• If determinant = 0 → Rank < order
• Rank remains unchanged under elementary row operations

Methods to Find Rank

1. Using Determinant (for square matrix)

If |A| ≠ 0 → Rank = n (full rank)

2. Using Echelon Form (Row Reduction)

Convert matrix to row echelon form and count non-zero rows.

3. Using Minors

Rank is the highest order of non-zero minor.

Row Echelon Form

• All zero rows at bottom
• Leading element (pivot) moves right
• Elements below pivot are zero

Solved Examples

Example 1

Find rank of:

|A| = (1×4 − 2×3) = −2 ≠ 0

Rank = 2

Example 2

Find rank of:

Second row = 2 × first row → rows dependent

Rank = 1

Example 3

Find rank using row reduction:

R₂ → R₂ − 2R₁ → (0 0 0)
R₃ → R₃ − R₁ → (0 −1 −2)

Non-zero rows = 2

Rank = 2

Example 4

If determinant = 0 for 3×3 matrix, what is rank?

Rank ≤ 2

Shortcut Tricks

• Check determinant first (for square matrix)
• Look for proportional rows
• Use row reduction quickly
• Count non-zero rows in echelon form

Questions

Q1. Maximum rank of 3×4 matrix?

Answer: 3

Q2. Rank of identity matrix of order n?

Answer: n

Exam Tips

• Always simplify matrix first
• Use row operations smartly
• Avoid unnecessary determinant expansion
• Practice PYQs

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