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

# Determinant identity proof

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2074 Asoj, 5 marks
- 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-seven
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Use properties of determinants to show that [x² x²−(y−z)² yz; y² y²−(z−x)² zx; z² z²−(x−y)² xy] (determinant) = (x − y)(y − z)(z − x)(x + y + z)(x² + y² + z²).

## Short answer

By C₂→C₂−C₁, C₂→C₂+2C₃, C₂→C₂+C₁, factor (x²+y²+z²), then row operations and expansion, the determinant equals (x−y)(y−z)(z−x)(x+y+z)(x²+y²+z²).

## Answer

Let

```text
Δ = | x²  x²−(y−z)²  yz |
    | y²  y²−(z−x)²  zx |
    | z²  z²−(x−y)²  xy |
```

**Step 1: C₂ → C₂ − C₁.** The second column becomes −(y−z)², −(z−x)², −(x−y)². Take (−1) common:

```text
Δ = −| x²  (y−z)²  yz |
     | y²  (z−x)²  zx |
     | z²  (x−y)²  xy |
```

**Step 2: C₂ → C₂ + 2C₃.** Since (y−z)² + 2yz = y² + z², etc.:

```text
Δ = −| x²  y²+z²  yz |
     | y²  z²+x²  zx |
     | z²  x²+y²  xy |
```

**Step 3: C₂ → C₂ + C₁.** Every entry of the middle column becomes x² + y² + z². Take it common:

```text
Δ = −(x²+y²+z²) | x²  1  yz |
                | y²  1  zx |
                | z²  1  xy |
```

**Step 4: R₁ → R₁ − R₂, R₂ → R₂ − R₃.**

- R₁: (x² − y², 0, yz − zx) = ((x−y)(x+y), 0, −z(x−y))
- R₂: (y² − z², 0, zx − xy) = ((y−z)(y+z), 0, −x(y−z))

```text
Δ = −(x²+y²+z²) | (x−y)(x+y)   0   −z(x−y) |
                | (y−z)(y+z)   0   −x(y−z) |
                |     z²       1      xy    |
```

**Step 5: Expand along C₂.** The entry 1 is at (3, 2), sign (−1)⁵ = −1. Take (x−y) from R₁ and (y−z) from R₂:

```text
inner = −(x−y)(y−z) | x+y  −z |
                     | y+z  −x |
      = −(x−y)(y−z) [ −x(x+y) + z(y+z) ]
      = −(x−y)(y−z) [ (z² − x²) + y(z − x) ]
      = −(x−y)(y−z)(z−x)(x+y+z)
```

Therefore Δ = −(x²+y²+z²) · [−(x−y)(y−z)(z−x)(x+y+z)]

**Δ = (x − y)(y − z)(z − x)(x + y + z)(x² + y² + z²).** ∎

## Check yourself

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

1. What does C2 -> C2 - C1 do?
   A. Makes column 2 equal -(y-z)^2, -(z-x)^2, -(x-y)^2
   B. Makes column 2 zero
   C. Doubles column 2
   D. Swaps columns
   Answer: A. x^2 - (y-z)^2 - x^2 = -(y-z)^2, so -1 is taken common. (Key point: Step 1)

2. Why is C2 -> C2 + 2C3 used?
   A. To create a zero column
   B. Because (y-z)^2 + 2yz = y^2 + z^2
   C. To factor out x
   D. To make C3 constant
   Answer: B. It converts the squared differences into sums of squares. (Key point: Step 2)

3. After C2 -> C2 + C1, what is common in column 2?
   A. x + y + z
   B. xyz
   C. x^2 + y^2 + z^2
   D. 1
   Answer: C. Every entry becomes x^2 + y^2 + z^2. (Key point: Common factor)

4. Which is the final result?
   A. (x+y+z)(x^2+y^2+z^2)
   B. (x-y)(y-z)(z-x)(x^2+y^2+z^2)
   C. xyz(x+y+z)
   D. (x-y)(y-z)(z-x)(x+y+z)(x^2+y^2+z^2)
   Answer: D. The expansion yields the factors (x-y)(y-z)(z-x)(x+y+z) times (x^2+y^2+z^2). (Key point: Final form)

## Close variants set in other exams

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

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