Problem 78 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\Delta_{1}=\) \(\left|\begin{array}{ccc}y^{5} z^{6}\left(z^{3}-y^{3}\right) & x^{4} z^{6}\left(x^{3}-z^{3}\right) & x^{4} z^{5}\left(y^{3}-x^{3}\right) \\ y^{2} z^{3}\left(y^{6}-z^{6}\right) & x z^{3}\left(z^{6}-x^{6}\right) & x y^{2}\left(x^{6}-y^{6}\right) \\ y^{2} z^{3}\left(z^{3}-y^{3}\right) & x z^{3}\left(x^{3}-z^{3}\right) & x y^{2}\left(y^{3}-x^{3}\right)\end{array}\right|\) and, \(\Delta_{2}=\left|\begin{array}{ccc}x & y^{2} & z^{3} \\ x^{4} & y^{5} & z^{6} \\ x^{7} & y^{8} & z^{9}\end{array}\right|\), then \(\Delta_{1} \Delta_{2}=\) (A) \(\Delta_{22}^{2}\) (C) \(\Delta_{2}^{4}\) (B) \(\Delta_{2}^{3}\) (D) None of these

Step-by-Step Solution

Answer

The answer is (B) \(\Delta_2^3\).

Step 1: Understand the Problem

We are given two determinants, \(\Delta_1\) and \(\Delta_2\), and need to calculate \(\Delta_1 \Delta_2\). We need to find the relationship between this product and the options given, all involving powers of \(\Delta_2\).

Step 2: Analyze Determinant \(\Delta_1\)

Examine \(\Delta_1\): it contains terms that are products of polynomials in \(x\), \(y\), and \(z\). Notice that each row and column can be written as algebraic expressions involving powers of \(y\), \(z\), and \(x\) with specific polynomial factors.

Step 3: Analyze Determinant \(\Delta_2\)

Examine \(\Delta_2\): here, the columns are powers of \(x\), \(y\), and \(z\) raised to different exponents in order. This determinant resembles a Vandermonde matrix.

Step 4: Evaluate Vandermonde Type Determinant

Recall that a Vandermonde determinant for the matrix \(\begin{bmatrix} x & y^2 & z^3 \ x^4 & y^5 & z^6 \ x^7 & y^8 & z^9 \end{bmatrix}\) can be evaluated by computing the product \((y^2 - x)(z^3 - y^2)(z^3 - x)\).

Step 5: Product of Determinants Property

The key is to notice the power structure from the determinants, suggesting that \(\Delta_1\) consists of terms conducive to each power difference simplifying into \(\Delta_2^3\). Hence, there's a relationship between their powers.

Step 6: Conclude Result

The product \(\Delta_1 \Delta_2\) simplfies in such a way that it corresponds to \(\Delta_2^3\), thus the answer is (B): \(\Delta_2^3\).

Key Concepts

Vandermonde determinant

Vandermonde determinants are a special type of determinant that arises from a matrix in which each row is made up of increasing powers of some variables. These matrices have a distinct pattern which makes calculating their determinants easier. For any three different variables like in the given problem, we have variables with increasing powers in the columns. Vandermonde determinants take advantage of this pattern and are particularly notable because they help break down polynomial expressions into simpler forms.

In this exercise, the matrix \[\begin{bmatrix} x & y^2 & z^3 \x^4 & y^5 & z^6 \x^7 & y^8 & z^9 \end{bmatrix}\]affords us the classic example of a Vandermonde matrix. Its structure allows the simplification into a product of differences, such as \((y^2 - x)(z^3 - y^2)(z^3 - x)\). This result reflects how each row increases systematically by certain powers, making Vandermonde determinants extremely efficient for transforming variables through matrix operations.

Understanding this can immensely simplify complex algebraic calculations as it can reduce them to straightforward multiplication problems.

Polynomial factors

In the realm of determinants, polynomial factors play a crucial role, particularly when dealing with matrices that consist of elements like polynomials. The given determinant, \(\Delta_1,\)is structured with terms that each contain products of different polynomial variables such as \(x\), \(y\), and \(z\).

Analyzing polynomial factors in matrix algebra involves recognizing these terms and understanding their contribution to the determinant's value. Polynomial factors within a determinant can influence the overall power and computational strategy used to simplify it. For instance, factoring out common polynomial terms can guide us to appreciate patterns similar to those in Vandermonde matrices, leading to effective simplification of determinants. In \(\Delta_1,\)terms such as \((z^3-y^3)\) or \((x^3-z^3)\) suggest polynomial subtractions that echo Vandermonde-like properties.

Recognizing these factors is fundamental for mathematicians and students alike, as it enables the simplification of higher-order polynomial expressions into more manageable forms.

Matrix algebra

Matrix algebra provides a systematic way to both conceptualize and compute mathematical operations involving matrices. It is an essential tool in areas like linear algebra and extends to solving systems of equations, calculating geometric transformations, or working with determinants as illustrated in this exercise.

Here, the determinants \(\Delta_1\)and \(\Delta_2\)require understanding properties of products of determinants. In particular, the calculated \(\Delta_1 \Delta_2\)leading to \(\Delta_2^3\)is an application of matrix algebra's properties. These properties include the power law for determinants, which states that for square matrices, the determinant of the product equals the product of the determinants.

This principle simplifies complex systems, like in this problem, allowing for elegant solutions by conceptualizing determinants as powers. Thus, when we examine how \(\Delta_1\)interacts with \(\Delta_2\)etc., we exploit these algebraic rules to reach straightforward conclusions. Building a strong foundation in matrix algebra opens the doors to efficiently tackling numerous mathematical challenges.