---
title: "Rank of a matrix by reduction [1,2,1,2;1,3,2,2;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:rank-of-a-matrix-by-reduction-nine"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-nine
site: BCT Engineering
---

# Rank of a matrix by reduction [1,2,1,2;1,3,2,2;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2068 Baishakh, 5 marks
- Video: [Rank of the matrix (echelon form) problem part1. 5](https://www.youtube.com/watch?v=WxAXVYvM1o8) — Rajesh Lekkalu
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-nine
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the rank of the following matrix by reducing it to normal form: [1 2 1 2; 1 3 2 2; 2 4 3 4; 3 7 4 6].

## Short answer

Elementary row and column operations reduce the matrix to normal form [I3 0; 0 0], so rank = 3.

## Answer

The rank is the order of the identity block in the normal form [Iᵣ 0; 0 0].

```text
A = [1 2 1 2]
    [1 3 2 2]
    [2 4 3 4]
    [3 7 4 6]
```

**Step 1.** R₂ → R₂ − R₁, R₃ → R₃ − 2R₁, R₄ → R₄ − 3R₁:

```text
[1 2 1 2]
[0 1 1 0]
[0 0 1 0]
[0 1 1 0]
```

**Step 2.** R₄ → R₄ − R₂:

```text
[1 2 1 2]
[0 1 1 0]
[0 0 1 0]
[0 0 0 0]
```

**Step 3 (column operations).** C₂ → C₂ − 2C₁, C₃ → C₃ − C₁, C₄ → C₄ − 2C₁ clear the first row:

```text
[1 0 0 0]
[0 1 1 0]
[0 0 1 0]
[0 0 0 0]
```

Then C₃ → C₃ − C₂ clears the second row, and the entries of R₃ are already isolated:

```text
[1 0 0 0]
[0 1 0 0]   =  [I₃ 0]
[0 0 1 0]      [0  0]
[0 0 0 0]
```

The normal form contains I₃.

**Answer: rank(A) = 3.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In normal form [Iᵣ 0; 0 0], the rank is:
   A. The number of zero rows
   B. The order r of the identity block
   C. The number of columns
   D. The determinant
   Answer: B. Rank equals the order of the identity block. (Key point: Definition of rank)

2. After R₂−R₁, R₃−2R₁, R₄−3R₁, row 3 is:
   A. (0,0,1,0)
   B. (0,1,1,0)
   C. (0,0,0,0)
   D. (0,1,2,0)
   Answer: A. R₃ = (2,4,3,4) − 2(1,2,1,2) = (0,0,1,0). (Key point: Step 1)

3. Why does R₄ become zero?
   A. R₄ was already zero
   B. Column operations remove it
   C. R₄ is divided by 0
   D. R₄ − R₂ makes (0,1,1,0) − (0,1,1,0) = 0
   Answer: D. After step 1 R₂ and R₄ are equal, so R₄ − R₂ = 0. (Key point: Zero row)

4. What is the rank?
   A. 2
   B. 4
   C. 3
   D. 1
   Answer: C. The normal form contains I₃, so rank = 3. (Key point: Answer)

## Close variants set in other exams

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