---
title: "Rank of a matrix by reduction [1,−1,−2,−4;2,3,−1,−1;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:rank-of-a-matrix-by-reduction-eleven"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-eleven
site: BCT Engineering
---

# Rank of a matrix by reduction [1,−1,−2,−4;2,3,−1,−1;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2072 Kartik, 5 marks
- Video: [Find the Rank of Matrix [[2,3,-1,-1],[1,-1,-2,-4],[3,1,3,-2],[6,3,0,-7]] |VTU M1/M2 B.E. ](https://www.youtube.com/watch?v=GeoFeQJ1EEo) — SK-Academy Of Mathematics 
- 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-eleven
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the rank of the matrix [1 −1 −2 −4; 2 3 −1 −1; 3 1 3 −2; 6 3 0 −7].

## Short answer

After R2−2R1, R3−3R1, R4−6R1 and then 5R3−4R2, 5R4−9R2, the matrix reduces to three non-zero rows with one zero row, so rank = 3.

## Answer

```text
A = [1 −1 −2 −4]
    [2  3 −1 −1]
    [3  1  3 −2]
    [6  3  0 −7]
```

**Step 1.** R₂ → R₂ − 2R₁, R₃ → R₃ − 3R₁, R₄ → R₄ − 6R₁:

```text
[1 −1 −2 −4]
[0  5  3  7]
[0  4  9 10]
[0  9 12 17]
```

**Step 2.** R₃ → 5R₃ − 4R₂, R₄ → 5R₄ − 9R₂:

```text
R₃: (0, 20−20, 45−12, 50−28) = (0, 0, 33, 22)
R₄: (0, 45−45, 60−27, 85−63) = (0, 0, 33, 22)

[1 −1 −2 −4]
[0  5  3  7]
[0  0 33 22]
[0  0 33 22]
```

**Step 3.** R₄ → R₄ − R₃:

```text
[1 −1 −2 −4]
[0  5  3  7]
[0  0 33 22]
[0  0  0  0]
```

Three non-zero rows remain, so the matrix is of rank 3 (|A| = 0).

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

## Check yourself

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

1. After clearing column 1, row 2 is:
   A. (0,3,5,7)
   B. (0,5,3,7)
   C. (0,−5,3,7)
   D. (0,5,−3,7)
   Answer: B. R₂ − 2R₁ = (0,5,3,7). (Key point: Step 1)

2. R₃→5R₃−4R₂ and R₄→5R₄−9R₂ give:
   A. Identical rows (0,0,33,22)
   B. Zero rows
   C. (0,0,1,0) and (0,0,0,1)
   D. Different non-zero rows
   Answer: A. Both become (0,0,33,22). (Key point: Elimination)

3. Why is the last row zero?
   A. R₄ was zero
   B. R₁ was deleted
   C. rank is 4
   D. R₄−R₃ = 0 since the rows are equal
   Answer: D. Equal rows subtract to zero. (Key point: Zero row)

4. The rank is:
   A. 4
   B. 2
   C. 3
   D. 1
   Answer: C. Three non-zero rows remain. (Key point: Rank)

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