Test: Matrices & Determinants - 5 - Mathematics MCQ

Problem Statement:

We are given a table with values of a, b, and c, and their corresponding cubes a³, b³, and c³. The expression provided is:

λ(a - b)(b - c)(c - a)

We need to determine the value of λ.

Understanding the Problem:

The problem involves a determinant or a similar expression that simplifies to the form λ(a - b)(b - c)(c - a). Our goal is to find λ.

Approach to the Solution:

1. Identify the Structure:

   • The expression (a - b)(b - c)(c - a) appears in many determinant-based formulas, particularly Vandermonde determinants.

2. Consider the Given Table:

   • The table suggests a relationship between a, b, c, and their cubes. This often corresponds to determinant properties.

3. Use Known Identities:

   • The determinant of a Vandermonde-like matrix often results in expressions involving (a - b)(b - c)(c - a).

4. Calculate the Determinant:

   • If we consider the table as a matrix, we compute its determinant and compare it to the given expression to find λ.

Detailed Calculations:

The given determinant is:

| 1 a a³ |
| 1 b b³ |
| 1 c c³ |

Using determinant expansion and known properties, the determinant simplifies to:

(a - b)(b - c)(c - a)(a + b + c)

Comparing this result with the given expression λ(a - b)(b - c)(c - a), we can see that:

λ = a + b + c

Final Answer:

λ = a + b + c