---
title: "Rank of a matrix by reduction [2,−2,0,6;4,2,0,2;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:rank-of-a-matrix-by-reduction"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction
site: BCT Engineering
---

# Rank of a matrix by reduction [2,−2,0,6;4,2,0,2;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2082 Shrawan, 5 marks; 2076 Asoj, 5 marks; 2067 Asar (asked 3 times)
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the rank of the matrix [2 −2 0 6; 4 2 0 2; 1 −1 0 3; 1 −2 1 2] reducing it into echelon form.

## Short answer

Reduce to row echelon form: [1 −1 0 3; 0 −1 1 −1; 0 0 6 −16; 0 0 0 0]. There are three non-zero rows, so rank = 3.

## Answer

The rank of a matrix is the number of non-zero rows in its row echelon form.

```text
A = [2 −2 0 6]
    [4  2 0 2]
    [1 −1 0 3]
    [1 −2 1 2]
```

**Step 1.** R₁ ↔ R₃ to get a leading 1:

```text
[1 −1 0  3]
[4  2 0  2]
[2 −2 0  6]
[1 −2 1  2]
```

**Step 2.** R₂ → R₂ − 4R₁, R₃ → R₃ − 2R₁, R₄ → R₄ − R₁:

```text
[1 −1 0   3]
[0  6 0 −10]
[0  0 0   0]
[0 −1 1  −1]
```

**Step 3.** Interchange R₂ and R₄ (and move the zero row to the bottom):

```text
[1 −1 0   3]
[0 −1 1  −1]
[0  6 0 −10]
[0  0 0   0]
```

**Step 4.** R₃ → R₃ + 6R₂:

```text
[1 −1 0   3]
[0 −1 1  −1]
[0  0 6 −16]
[0  0 0   0]
```

This is in echelon form with 3 non-zero rows.

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

(Check: in the original matrix R₁ = 2R₃, which explains the zero row, and the three pivots show rank = 3.)

## Check yourself

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

1. How is rank read from an echelon form?
   A. Count the non-zero rows
   B. Count the zero rows
   C. Add the diagonal entries
   D. Count the columns
   Answer: A. Rank equals the number of non-zero rows in echelon form. (Key point: Rank from echelon form)

2. Why does a zero row appear in the reduction?
   A. Because R2 = R4
   B. Because the last column is zero
   C. In the original matrix R1 = 2R3, so R1 - 2R3 = 0
   D. Because the matrix is symmetric
   Answer: C. Row 1 is a multiple of row 3, so one row becomes zero. (Key point: Zero row cause)

3. What is the rank?
   A. 4
   B. 3
   C. 2
   D. 1
   Answer: B. Three non-zero rows remain. (Key point: Result)

## Close variants set in other exams

- [Rank of a matrix by reduction [1,3,−2,1;1,1,1,1;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-two) — Mathematics III, chapter 1, asked 14×
- [Rank of a matrix by reduction [1,0,−5,6;3,−2,1,2;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-three) — Mathematics III, chapter 1, asked 14×
- [Rank of a matrix by reduction [3,1,4;0,5,8;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-four) — Mathematics III, chapter 1, asked 14×
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