---
title: "Determinant identity proof"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:determinant-identity-proof"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/determinant-identity-proof
site: BCT Engineering
---

# Determinant identity proof

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2081 Bhadra, 5 marks; 2080 Bhadra, 5 marks (asked 2 times)
- Video: [Linear Algebra : Proofs for Determinant Identities](https://www.youtube.com/watch?v=jcb3o4MITgg) — Maths Resource
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/determinant-identity-proof
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Using properties of determinants, show that [a b c; b+c c+a a+b; a² b² c²] (determinant) = (a − b)(b − c)(a − c)(a + b + c).

## Short answer

Perform R₂→R₂+R₁ and factor a+b+c, then column operations and expansion along a row with zeros yield (a−b)(b−c)(a−c)(a+b+c).

## Answer

Let

```text
Δ = |  a    b    c  |
    | b+c  c+a  a+b |
    |  a²   b²   c² |
```

**Step 1: R₂ → R₂ + R₁.** Every entry of the new second row becomes a + b + c:

```text
Δ = |  a    b    c  |
    | s    s    s  |      where s = a + b + c
    |  a²   b²   c² |
```

Take s common from R₂:

```text
Δ = s | a   b   c  |
      | 1   1   1  |
      | a²  b²  c² |
```

**Step 2: C₁ → C₁ − C₂, C₂ → C₂ − C₃.**

```text
Δ = s | a−b    b−c    c  |
      |  0      0     1  |
      | a²−b²  b²−c²  c² |
```

**Step 3: Expand along R₂.** The only non-zero entry is 1 in position (2, 3), with sign (−1)^(2+3) = −1:

```text
Δ = −s [ (a−b)(b²−c²) − (b−c)(a²−b²) ]
  = −s (a−b)(b−c) [ (b+c) − (a+b) ]
  = −s (a−b)(b−c)(c−a)
  = s (a−b)(b−c)(a−c)
```

**Therefore Δ = (a − b)(b − c)(a − c)(a + b + c).** ∎

## Check yourself

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

1. Which operation makes row 2 identical entries?
   A. R2 -> R2 + R1
   B. R2 -> R2 - R1
   C. C1 -> C1 + C2
   D. R3 -> R3 - R1
   Answer: A. Each entry becomes a + b + c. (Key point: First operation)

2. What factor is taken from R2?
   A. abc
   B. a - b
   C. a + b + c
   D. 1
   Answer: C. Row 2 then becomes (1,1,1). (Key point: Factor)

3. Why is expansion along R2 easy after C1->C1-C2, C2->C2-C3?
   A. R2 becomes (1, 1, 1)
   B. R2 becomes (0, 0, 1)
   C. R2 becomes zero
   D. R2 becomes (a, b, c)
   Answer: B. Only one non-zero entry. (Key point: Expansion)

4. The determinant equals
   A. (a+b)(b+c)(c+a)
   B. abc(a+b+c)
   C. (a-b)(b-c)(c-a)
   D. (a-b)(b-c)(a-c)(a+b+c)
   Answer: D. Using -(c-a) = (a-c). (Key point: Result)

## Close variants set in other exams

- [Determinant identity: 2(a+b+c)³](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-2-a-b-c) — Mathematics III, chapter 1, asked 6×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-two) — Mathematics III, chapter 1, asked 4×
- [Determinant identity: (a + b + c)³](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-a-b-c-3) — Mathematics III, chapter 1, asked 6×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-seven) — Mathematics III, chapter 1, asked 6×

## Related in Mathematics III

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