---
title: "Determinant identity: −(b − a)⁴"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:determinant-identity-b-a"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/determinant-identity-b-a
site: BCT Engineering
---

# Determinant identity: −(b − a)⁴

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2079 Baishakh, 5 marks; 2070 Asar, 5 marks; 2067 Magh (asked 3 times)
- 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-b-a
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Applying properties of determinants, prove that [a b a a; a b b b; b b b a; a a b a] (determinant) = −(b − a)⁴.

## Short answer

Using row operations, factoring d = b−a, and expanding, the determinant equals d³(−d) = −d⁴ = −(b−a)⁴.

## Answer

Let

```text
Δ = | a b a a |
    | a b b b |
    | b b b a |
    | a a b a |
```

and put d = b − a.

#### Step 1: Make zeros in the first column
Apply R₂ → R₂ − R₁, R₃ → R₃ − R₁, R₄ → R₄ − R₁. Then R₂ − R₁ = (0, 0, d, d), R₃ − R₁ = (d, 0, d, 0) and R₄ − R₁ = (0, −d, d, 0):

```text
Δ = | a   b    a    a  |
    | 0   0    d    d  |
    | d   0    d    0  |
    | 0  −d    d    0  |
```

#### Step 2: Take d common from R₂, R₃, R₄
```text
Δ = d³ | a  b  a  a |
       | 0  0  1  1 |
       | 1  0  1  0 |
       | 0 −1  1  0 |
```

#### Step 3: C₃ → C₃ − C₄
```text
Δ = d³ | a  b  0  a |
       | 0  0  0  1 |
       | 1  0  1  0 |
       | 0 −1  1  0 |
```

#### Step 4: Expand along R₂
The only non-zero entry is 1 at position (2, 4), with sign (−1)^(2+4) = +1:

```text
Δ = d³ · | a  b  0 |
         | 1  0  1 |
         | 0 −1  1 |
```

Expanding this 3 × 3 determinant along R₁:

```text
a(0·1 − 1·(−1)) − b(1·1 − 0) + 0 = a − b = −d
```

So Δ = d³(−d) = −d⁴ = **−(b − a)⁴**. ∎

## Check yourself

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

1. What first step creates zeros in column 1?
   A. C1 -> C1 + C2
   B. Swap R1 and R4
   C. R2, R3, R4 -> each minus R1
   D. R1 -> R1 - R2
   Answer: C. Subtracting R1 gives rows (0,0,d,d), (d,0,d,0), (0,-d,d,0) with d = b - a. (Key point: First step)

2. What is taken common from R2, R3, R4?
   A. a, giving a^3
   B. d = b - a, giving d^3
   C. d^2
   D. (a + b)
   Answer: B. Each of the three rows has d common. (Key point: Factor)

3. Which operation makes row 2 have a single non-zero entry?
   A. C3 -> C3 - C4
   B. C1 -> C1 - C2
   C. C2 -> C2 + C4
   D. R2 -> 2R2
   Answer: A. It turns row 2 into (0,0,0,1). (Key point: Column operation)

4. What is the determinant?
   A. (b-a)^4
   B. (b-a)^3
   C. -(b-a)^3
   D. -(b-a)^4
   Answer: D. d^3 times (a - b) = d^3(-d) = -d^4. (Key point: Result)

## Close variants set in other exams

- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-three) — Mathematics III, chapter 1, asked 9×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-five) — Mathematics III, chapter 1, asked 9×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-13) — Mathematics III, chapter 1, asked 9×
- [Determinant identity proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-14) — Mathematics III, chapter 1, asked 9×

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