Problem 52 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant \(\left|\begin{array}{ccc}\left(a-a_{1}\right)^{-2} & \left(a-a_{1}\right)^{-1} & a_{1}^{-1} \\ \left(a-a_{2}\right)^{-2} & \left(a-a_{2}\right)^{-1} & a_{2}^{-1} \\ \left(a-a_{3}\right)^{-2} & \left(a-a_{3}\right)^{-1} & a_{3}^{-1}\end{array}\right|\) is (A) \(\frac{a^{2} \Pi\left(a_{i}-a_{j}\right)}{\pi a_{i} \Pi\left(a-a_{i}\right)^{2}}\) (B) \(\frac{-a^{2} \Pi\left(a_{i}-a_{j}\right)}{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}\) (C) \(\frac{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}{a^{2} \Pi\left(a_{i}-a_{j}\right)}\) (D) \(-\frac{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}{a^{2} \Pi\left(a_{i}-a_{j}\right)}\)

Step-by-Step Solution

Answer

Option (B) is correct.

Step 1: Analyze the Problem

We need to find the determinant of a matrix with the elements defined in terms of the variable \(a\) and \(a_i\). The matrix has three rows and three columns. The determinant calculation involves evaluating \( \left| \begin{array}{ccc} (a-a_1)^{-2} & (a-a_1)^{-1} & a_1^{-1} \ (a-a_2)^{-2} & (a-a_2)^{-1} & a_2^{-1} \ (a-a_3)^{-2} & (a-a_3)^{-1} & a_3^{-1} \end{array} \right| \).

Step 2: Identify Patterns or Simplifications

Notice the structure of the matrix. Each row has elements with similar formats, where each entry involves \((a-a_i)^{-n}\) and \(a_i^{-1}\). We will try row operations or column operations if possible to simplify the determinant.

Step 3: Apply Elementary Row Operations

Subtract the first column times \( (a - a_1) \) from the second column and the second column times \( (a - a_2) \) from the third column. This reduces the second and third columns into simpler expressions.

Step 4: Calculate the Determinant through Expansion

Once simplification is made via elementary operations, calculate the determinant by expanding along the first row. It should result in a simplified expression that allows you to match with one of the options.

Step 5: Verify with Options

Compare the result obtained from calculating the determinant with the given options: (A), (B), (C), and (D). The structure of the simplified determinant expression should correspond with the negative sign seen in Option (B).

Key Concepts

Elementary Row Operations

Elementary row operations are key tools for manipulating matrices. They help us simplify a matrix, making the calculation of its determinant much easier. These operations include:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting a multiple of one row to another

In the context of determinants, it’s important to note that these operations can change the value of the determinant. For instance, swapping two rows changes the sign of the determinant. Multiplying a row by a factor also multiplies the determinant by the same factor.

In the original exercise, a key operation used was subtracting a multiple of one row from another. This step simplified the matrix structure, leading to a more straightforward determinant calculation. With skillful use of elementary row operations, we systematically reduce the complexity of matrix expressions.

Matrix Simplification

Matrix simplification involves using techniques like row operations to make a matrix easier to work with, especially when calculating determinants. Consider that the primary goal is to obtain as many zeroes as possible in the matrix in order to streamline the calculations.

In the original exercise, simplification was achieved by systematically transforming the matrix through operations that maintain its determinant properties. By focusing on transforming elements into simpler or more recognizable forms, one can get closer to a result that is either easily expandable or comparable to known outcomes.
To apply this in practice, identify patterns. As shown, each row had elements in a specific form, like \((a-a_i)^{-n}\). Recognizing such patterns allows targeted operations to transform the matrix efficiently. Thus, breaking down complex operations into digestible and strategic actions increases problem-solving efficiency.

Determinant Expansion

Determinant expansion or cofactor expansion is a method used to compute the determinant of a matrix. It involves breaking down a large determinant into smaller ones, making it easier to manage. The expansion is usually done along a chosen row or column.

The basic formula for expanding a determinant for a 3x3 matrix along the first row is:\[|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\]where \(a_{ij}\) are elements of the matrix and \(C_{ij}\) the corresponding cofactors.
In the original solution, after simplifying the matrix, the determinant was expanded along the first row. This technique breaks down the determinant into smaller, manageable parts, leveraging the simpler expressions obtained from the prior simplification processes. By doing so, you transform the problem into a series of smaller calculations leading to your final answer.
Remember, choosing which row or column to expand upon can be strategic. You typically want to select one with the most zeroes or simplest elements, making further calculations less laborious.