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

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

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2073 Shrawan, 5 marks; 2070 Chaitra, 5 marks; 2069 Asar, 5 marks (asked 3 times)
- Video: [Finding the Rank of a Matrix | Solved Example Pt.1 | Linear Algebra](https://www.youtube.com/watch?v=mrvZZGb0830) — Mathematics Tutor
- 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Find the rank of the matrix [1 3 −2 1; 1 1 1 1; 2 0 −3 2; 3 3 −3 3] by reducing it into normal form.

## Short answer

The matrix reduces to echelon form with three non-zero rows, and its normal form contains the 3×3 identity block. Hence rank(A) = 3.

## Answer

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

```text
A = [1  3 −2 1]
    [1  1  1 1]
    [2  0 −3 2]
    [3  3 −3 3]
```

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

```text
[1  3 −2 1]
[0 −2  3 0]
[0 −6  1 0]
[0 −6  3 0]
```

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

```text
[1  3 −2 1]
[0 −2  3 0]
[0  0 −8 0]
[0  0 −6 0]
```

**Step 3.** R₄ → R₄ − (3/4)R₃:

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

**Step 4 (to normal form).** Use column operations. C₂ → C₂ − 3C₁, C₃ → C₃ + 2C₁ and C₄ → C₄ − C₁ clear the first row, leaving the rows (1, 0, 0, 0), (0, −2, 3, 0), (0, 0, −8, 0), (0, 0, 0, 0). Then C₃ → C₃ + (3/2)C₂ clears the 3 in the second row. Finally, R₂ → (−1/2)R₂ and R₃ → (−1/8)R₃ make the pivots 1, and the matrix becomes

```text
[I₃ 0]
[0  0]
```

The normal form contains the 3 × 3 identity matrix, so

**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 [1,0,−5,6;3,−2,1,2;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-three) — 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

- [Rank of a matrix by reduction [1,0,−5,6;3,−2,1,2;…]](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Arank-of-a-matrix-by-reduction-seven) — Mathematics III, chapter 1, asked 2×
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