---
title: "Rank of a matrix by reduction [3,1,4;0,5,8;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:rank-of-a-matrix-by-reduction-six"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-six
site: BCT Engineering
---

# Rank of a matrix by reduction [3,1,4;0,5,8;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2080 Bhadra, 1+4 marks
- Video: [Rank of matrix | normal form method | Matrix | PART 2 | #barunmaths #rankofmatrix](https://www.youtube.com/watch?v=GKcIzkiK5ew) — Dr Barun Kumar yadav
- 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-six
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Define the rank of a matrix. Find the rank of the matrix [3 1 4; 0 5 8; −3 4 4; 1 2 4] by reducing it to normal form.

## Short answer

Rank is the number of non-zero rows in echelon form. Row reduction yields echelon form [1 2 4; 0 5 8; 0 0 0; 0 0 0] with two non-zero rows. Therefore rank(A) = 2.

## Answer

The **rank** of a matrix is the order of its largest non-zero minor; it is also the number of non-zero rows in the echelon form, or the size of the identity block in the normal form [Iᵣ 0; 0 0].

```text
A = [ 3 1 4]
    [ 0 5 8]
    [−3 4 4]
    [ 1 2 4]
```

**Step 1.** R₁ ↔ R₄ to get a pivot 1:

```text
[ 1 2 4]
[ 0 5 8]
[−3 4 4]
[ 3 1 4]
```

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

```text
[1   2   4]
[0   5   8]
[0  10  16]
[0  −5  −8]
```

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

```text
[1  2  4]
[0  5  8]
[0  0  0]
[0  0  0]
```

Two non-zero rows remain. Further column operations (C₂ → C₂ − 2C₁, C₃ → C₃ − 4C₁, then C₃ → C₃ − (8/5)C₂ and scaling) give the normal form

```text
[I₂ 0]
[0  0]
```

**Answer: rank(A) = 2.**

(Check: the 3 × 3 minors are all zero, e.g. |3 1 4; 0 5 8; −3 4 4| = 3(20−32) − 1(0+24) + 4(0+15) = −36 − 24 + 60 = 0, while a 2 × 2 minor such as |1 2; 0 5| = 5 ≠ 0.)

## Check yourself

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

1. What is the aim of row operations here?
   A. Reduce to echelon form and count non-zero rows
   B. Make all entries 1
   C. Find the determinant
   D. Transpose the matrix
   Answer: A. Non-zero rows in the echelon form give the rank. (Key point: Using row operations)

2. What is the normal form of a matrix of rank r?
   A. [0 I_r; 0 0]
   B. [I_r 0; 0 0]
   C. [I_r I_r; 0 0]
   D. A diagonal of the original entries
   Answer: B. The normal form has an r x r identity block and zeros elsewhere. (Key point: Normal form)

3. What is the rank of this 4 x 3 matrix?
   A. 1
   B. 3
   C. 2
   D. 4
   Answer: C. The reduction leaves 2 non-zero rows, so the rank is 2. (Key point: Result)

4. Rank is the order of the
   A. smallest non-zero minor
   B. largest entry
   C. number of columns
   D. largest non-zero minor
   Answer: D. That is the minor definition of rank. (Key point: Definition)

## Close variants set in other exams

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