---
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-five"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-five
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: 2081 Baishakh, 1+4 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-five
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Define the rank of a matrix. Reduce the matrix [2 −2 0 6; 4 2 0 2; 1 −1 0 3; 1 −2 1 2] into echelon form and hence find its rank.

## Short answer

Rank is the number of non-zero rows in row echelon form. After row operations (swap rows, eliminate below pivots) the matrix reduces to echelon form with three non-zero rows. Hence rank = 3.

## Answer

The **rank** of a matrix is the order of its largest square sub-matrix with non-zero determinant. Equivalently, it 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. Which is a correct definition of rank?
   A. Number of rows of the matrix
   B. Number of zero rows in echelon form
   C. Sum of the diagonal entries
   D. Order of the largest square submatrix with non-zero determinant
   Answer: D. Rank is the order of the largest non-zero minor. (Key point: Definition of rank)

2. 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)

3. Why does a zero row appear in the reduction?
   A. In the original matrix R1 = 2R3
   B. R2 = R4
   C. The matrix has a zero column
   D. The determinant is 1
   Answer: A. Row 1 is twice row 3. (Key point: Zero row cause)

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

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