---
title: "Rank of a matrix by reduction [1,−1,2,−3;4,1,0,2;…]"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:rank-of-a-matrix-by-reduction-ten"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/rank-of-a-matrix-by-reduction-ten
site: BCT Engineering
---

# Rank of a matrix by reduction [1,−1,2,−3;4,1,0,2;…]

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2067 Magh
- 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-ten
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Reducing to normal form, find the rank of the matrix [1 −1 2 −3; 4 1 0 2; 0 3 0 4; 0 1 0 2].

## Short answer

Row operations reduce the matrix to upper triangular form with diagonal entries 1, 1, −8, −2, so the determinant is 16 ≠ 0, hence the rank is 4.

## Answer

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

**Step 1.** R₂ → R₂ − 4R₁:

```text
[1 −1  2 −3]
[0  5 −8 14]
[0  3  0  4]
[0  1  0  2]
```

**Step 2.** Interchange R₂ and R₄ to get a pivot 1:

```text
[1 −1  2 −3]
[0  1  0  2]
[0  3  0  4]
[0  5 −8 14]
```

**Step 3.** R₃ → R₃ − 3R₂, R₄ → R₄ − 5R₂:

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

**Step 4.** Interchange R₃ and R₄:

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

This is upper triangular with four non-zero diagonal entries (1, 1, −8, −2). Column operations (clearing the entries above each pivot) and then scaling reduce it to the normal form

```text
[I₄]   (the 4 × 4 identity matrix)
```

Since |A| = 1·1·(−8)·(−2) = 16 ≠ 0, A is non-singular.

**Answer: rank(A) = 4.**

## Check yourself

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

1. Why interchange R₂ and R₄ after step 1?
   A. To change the rank
   B. To get a pivot 1 in the second row
   C. To make the determinant zero
   D. To clear column 1
   Answer: B. R₄ = (0,1,0,2) provides a convenient pivot 1. (Key point: Pivot choice)

2. Final row-echelon diagonal entries are:
   A. 1, 1, −8, −2
   B. 1, 5, 0, 2
   C. 1, 1, 0, 0
   D. 1, 3, 4, 0
   Answer: A. The upper triangular form has diagonals 1, 1, −8, −2. (Key point: Triangular form)

3. What is |A| from the diagonal?
   A. −16
   B. 0
   C. 32
   D. 16
   Answer: D. 1·1·(−8)·(−2) = 16 (two row swaps cancel in sign), non-zero. (Key point: Determinant)

4. Rank of A:
   A. 3
   B. 2
   C. 4
   D. 1
   Answer: C. Four non-zero pivots, so rank 4 and A is non-singular. (Key point: Rank)

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