---
title: "Determinant identity: 2(a + b + c)³"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:determinant-identity-2-a-b-c-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/determinant-identity-2-a-b-c-two
site: BCT Engineering
---

# Determinant identity: 2(a + b + c)³

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2073 Chaitra, 5 marks
- Video: [Show that: det [[a - b - c, 2a, 2a], [2b, b - c - a, 2b], [2c, 2c, c - a - b]] = (a + b + c)^3](https://www.youtube.com/watch?v=cwJx3yqzVAA) — Mathematics Untold
- 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-2-a-b-c-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Distinguish a matrix and a determinant. Use the property of determinants to prove: [a+b+2c a b; c b+c+2a b; c a c+a+2b] (determinant) = 2(a + b + c)³.

## Short answer

A matrix is an array of numbers, while a determinant is a scalar computed from a square matrix. By C1 → C1+C2+C3, factor 2(a+b+c), then R2−R1 and R3−R1, the determinant becomes 2(a+b+c)·1·( a+b+c)·( a+b+c) = 2(a+b+c)³.

## Answer

#### Difference between a matrix and a determinant
| Matrix | Determinant |
|---|---|
| An arrangement (array) of numbers in rows and columns, written in brackets [ ] | A single number (value) calculated from a square array, written between bars \| \| |
| Can be of any order m × n | Defined only for a square array (n × n) |
| Has no numerical value of its own | Has a definite value |
| Interchanging rows and columns gives a different matrix (the transpose) | Interchanging rows and columns does not change the value |
| Multiplying by k multiplies every element | Multiplying by k multiplies only one row (or column) |

#### Proof
Let s = a + b + c and

```text
Δ = | a+b+2c      a        b    |
    |   c       b+c+2a     b    |
    |   c         a      c+a+2b |
```

**Step 1: C₁ → C₁ + C₂ + C₃.** Each entry of the new first column is 2(a + b + c):

- Row 1: (a + b + 2c) + a + b = 2a + 2b + 2c
- Row 2: c + (b + c + 2a) + b = 2a + 2b + 2c
- Row 3: c + a + (c + a + 2b) = 2a + 2b + 2c

```text
Δ = | 2s      a        b    |
    | 2s   b+c+2a      b    |
    | 2s      a      c+a+2b |
```

**Step 2:** Take 2s common from C₁:

```text
Δ = 2s | 1      a        b    |
       | 1   b+c+2a      b    |
       | 1      a      c+a+2b |
```

**Step 3: R₂ → R₂ − R₁, R₃ → R₃ − R₁.**

- R₂ − R₁ = (0, b + c + 2a − a, 0) = (0, s, 0)
- R₃ − R₁ = (0, 0, c + a + 2b − b) = (0, 0, s)

```text
Δ = 2s | 1  a  b |
       | 0  s  0 |
       | 0  0  s |
```

This is upper triangular, so the determinant is the product of the diagonal: 1 · s · s = s².

Δ = 2s · s² = **2(a + b + c)³.** ∎

## Check yourself

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

1. Which is true of a determinant but not a matrix?
   A. It can be m × n
   B. It has a numerical value and exists only for square arrays
   C. Transposing changes it
   D. Multiplying by k multiplies every element
   Answer: B. A determinant is a single value for a square array. (Key point: Matrix vs determinant)

2. Which operation creates a common factor in the first column?
   A. C₁ → C₁+C₂+C₃
   B. R₁ → R₁+R₂
   C. C₁ → C₁−C₂
   D. C₂ → C₂+C₃
   Answer: A. Each entry becomes 2(a+b+c). (Key point: Column operation)

3. R₂−R₁ and R₃−R₁ give rows:
   A. (0,0,s) and (0,s,0)
   B. (1,s,0) and (1,0,s)
   C. (0,a,b) and (0,b,a)
   D. (0,s,0) and (0,0,s)
   Answer: D. R₂−R₁ = (0,s,0), R₃−R₁ = (0,0,s) with s=a+b+c. (Key point: After row operations)

4. Δ equals:
   A. (a+b+c)³
   B. 2(a+b+c)²
   C. 2(a+b+c)³
   D. (a+b+c)²
   Answer: C. 2s · (1·s·s) = 2s³. (Key point: Result)

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