---
title: "Rank of a matrix by reduction [1,0,−5,6;3,−2,1,2;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:rank-of-a-matrix-by-reduction-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-three
site: BCT Engineering
---

# Rank of a matrix by reduction [1,0,−5,6;3,−2,1,2;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2079 Bhadra, 5 marks; 2073 Chaitra, 5 marks (asked 2 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the rank of the matrix [1 0 −5 6; 3 −2 1 2; 3 −2 −9 14; 4 −2 −4 8] by reducing it into normal form.

## Short answer

After row reduction, three non-zero rows remain; the normal form is [I₃ 0; 0 0]. Thus rank(A) = 3.

## Answer

```text
A = [1  0 −5  6]
    [3 −2  1  2]
    [3 −2 −9 14]
    [4 −2 −4  8]
```

**Step 1.** R₂ → R₂ − 3R₁, R₃ → R₃ − 3R₁, R₄ → R₄ − 4R₁:

```text
[1  0  −5    6]
[0 −2  16  −16]
[0 −2   6   −4]
[0 −2  16  −16]
```

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

```text
[1  0  −5    6]
[0 −2  16  −16]
[0  0 −10   12]
[0  0   0    0]
```

This is an echelon form with three non-zero rows.

**Normal form.** Column operations (C₄ → C₄ − 6C₁, C₃ → C₃ + 5C₁, then clearing the remaining entries of rows 2 and 3 using the pivots −2 and −10 and dividing rows by the pivots) reduce the matrix to

```text
[I₃ 0]
[0  0]
```

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

## 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 4 matrix?
   A. 1
   B. 2
   C. 3
   D. 4
   Answer: C. The reduction leaves 3 non-zero rows, so the rank is 3. (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 [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×
- [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-six) — Mathematics III, chapter 1, asked 10×
- [Rank of a matrix by reduction [1,−1,2,−3;4,1,0,2;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-ten) — Mathematics III, chapter 1, asked 10×
- [Rank of a matrix by reduction [2,−2,0,6;4,2,0,2;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction) — Mathematics III, chapter 1, asked 11×
- [Rank of a matrix by reduction [6,1,3,8;4,2,6,−1;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-eight) — Mathematics III, chapter 1, asked 11×
- [Rank of a matrix by reduction [1,2,1,2;1,3,2,2;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-nine) — Mathematics III, chapter 1, asked 10×

## Related in Mathematics III

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