---
title: "Rank of a matrix by reduction [6,1,3,8;4,2,6,−1;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:rank-of-a-matrix-by-reduction-eight"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-eight
site: BCT Engineering
---

# Rank of a matrix by reduction [6,1,3,8;4,2,6,−1;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2069 Chaitra, 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-eight
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the rank of the matrix [6 1 3 8; 4 2 6 −1; 10 3 9 7; 16 4 12 15] reducing it into triangular form.

## Short answer

Reduce matrix to echelon form via row operations; the non-zero rows are [6 1 3 8] and [0 4 12 −19], giving rank 2.

## Answer

The rank is the number of non-zero rows left when the matrix is reduced to triangular (echelon) form.

```text
A = [ 6 1  3  8]
    [ 4 2  6 −1]
    [10 3  9  7]
    [16 4 12 15]
```

**Step 1.** Eliminate the first column using R₁ (pivot 6): R₂ → 3R₂ − 2R₁, R₃ → 3R₃ − 5R₁, R₄ → 3R₄ − 8R₁:

```text
R₂: (12−12, 6−2, 18−6, −3−16)  = (0, 4, 12, −19)
R₃: (30−30, 9−5, 27−15, 21−40) = (0, 4, 12, −19)
R₄: (48−48, 12−8, 36−24, 45−64)= (0, 4, 12, −19)

[6 1  3   8]
[0 4 12 −19]
[0 4 12 −19]
[0 4 12 −19]
```

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

```text
[6 1  3   8]
[0 4 12 −19]
[0 0  0   0]
[0 0  0   0]
```

The matrix is in triangular (echelon) form with two non-zero rows.

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

(Check: in the original matrix R₃ = R₁ + R₂ and R₄ = R₁ + R₃, so only R₁ and R₂ are independent.)

## Check yourself

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

1. Rank by triangular form is:
   A. The number of zero rows
   B. The number of non-zero rows in echelon form
   C. The number of columns
   D. The sum of the diagonal
   Answer: B. Non-zero rows count the independent rows. (Key point: Rank definition)

2. To clear column 1 using pivot 6 without fractions, R2 becomes:
   A. R2 - 2R1
   B. 3R2 - 2R1
   C. 2R2 - 3R1
   D. R2 + 6R1
   Answer: B. 3(4) - 2(6) = 0. (Key point: First step)

3. After step 1, rows 2, 3, 4 all equal:
   A. (0, 4, 12, -19)
   B. (0, 0, 0, 0)
   C. (6, 1, 3, 8)
   D. (0, 2, 6, -1)
   Answer: A. Computed entries are identical, so two rows vanish in step 2. (Key point: Result of step 1)

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

## Close variants set in other exams

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