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

# Determinant identity: 2abc(a + b + c)³

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2078 Bhadra, 5 marks
- 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-2abc-a-b-c-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Use the properties of determinants to show that [(a+b)² ca bc; ca (b+c)² ab; bc ab (c+a)²] (determinant) = 2abc(a + b + c)³.

## Short answer

Multiply rows by c, a, b, take common factors from columns, then use column operations to factor (a+b+c)². Further reduction yields 2abc(a+b+c)³.

## Answer

Let

```text
Δ = | (a+b)²   ca      bc   |
    |  ca    (b+c)²    ab   |
    |  bc      ab    (c+a)² |
```

#### Step 1: Remove the products
Multiply R₁ by c, R₂ by a, R₃ by b. Then

```text
abc·Δ = | c(a+b)²   c²a      bc²   |
        |  ca²     a(b+c)²   a²b   |
        |  b²c      ab²    b(c+a)² |
```

Take c common from C₁, a common from C₂, b common from C₃. This gives a factor abc, which cancels with the abc on the left:

```text
Δ = | (a+b)²   c²      c²    |
    |  a²    (b+c)²    a²    |
    |  b²      b²    (c+a)²  |
```

To keep the algebra simple put x = c, y = a, z = b, so that a + b = y + z, b + c = z + x and c + a = x + y:

```text
Δ = | (y+z)²   x²      x²    |
    |  y²    (z+x)²    y²    |
    |  z²      z²    (x+y)²  |
```

Let s = x + y + z (= a + b + c).

#### Step 2: C₁ → C₁ − C₃, C₂ → C₂ − C₃
- Column 1: (y+z)² − x² = (y+z−x)s, 0, z² − (x+y)² = (z−x−y)s
- Column 2: 0, (z+x)² − y² = (z+x−y)s, z² − (x+y)² = (z−x−y)s

Take s from each of C₁, C₂ and write p = y+z−x, q = z+x−y, r = z−x−y:

```text
Δ = s² | p  0  x² |
       | 0  q  y² |
       | r  r  (x+y)² |
```

#### Step 3: R₃ → R₃ − R₁ − R₂
- r − p = −2y, r − q = −2x, (x+y)² − x² − y² = 2xy

```text
Δ = s² | p    0    x²  |
       | 0    q    y²  |
       | −2y −2x   2xy |
```

#### Step 4: Expand along R₁
```text
Δ/s² = p(2xyq + 2xy²) + x²(2yq)
     = 2xy [ p(q + y) + xq ]
```
With q + y = z + x: p(z + x) + xq = (y+z−x)(z+x) + x(z+x−y) = yz + xy + z² − x² + xz + x² − xy = z(x + y + z) = zs.

So Δ = s² · 2xy · zs = 2xyz s³, i.e.

**Δ = 2abc(a + b + c)³.** ∎

## Check yourself

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

1. What is the first step used to remove the products ca, bc, ab from the entries?
   A. Add all rows together
   B. Square every entry
   C. Multiply R1, R2, R3 by c, a, b and then take c, a, b out of C1, C2, C3
   D. Subtract R3 from R1
   Answer: C. The row multiplication gives a factor abc which is cancelled by taking c, a, b common from the columns. (Key point: Removing products)

2. With x = c, y = a, z = b, what does a + b become?
   A. x + y
   B. z + x
   C. x + y + z
   D. y + z
   Answer: D. a + b = y + z, b + c = z + x, c + a = x + y. (Key point: Substitution)

3. In C1 -> C1 - C3, the first entry (y+z)^2 - x^2 equals what?
   A. (y+z-x)(x+y+z)
   B. (y+z+x)^2
   C. (y+z)^2 - x
   D. y^2 + z^2 - x^2
   Answer: A. A^2 - B^2 = (A-B)(A+B) with A = y+z, B = x, so the factor s = x+y+z appears. (Key point: Difference of squares)

4. After the simplification the determinant equals what?
   A. abc(a+b+c)^2
   B. 2abc(a+b+c)^3
   C. (a+b+c)^3
   D. 4abc(a+b+c)^3
   Answer: B. Δ = s^2 · 2xy · zs = 2xyz s^3 = 2abc(a+b+c)^3. (Key point: Final result)

## 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 proof](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Adeterminant-identity-proof-seven) — Mathematics III, chapter 1, asked 6×

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