Problem 83 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\left\\{\Delta_{1}, \Delta_{2}, \Delta_{3}, \ldots, \Delta_{k}\right\\}\) be the set of third order determinants that can be made with the distinct non- zero real numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{9} .\) Then, (A) \(k=9 !\) (B) \(\sum_{i=1}^{k} \Delta_{i}=0\) (C) at least one \(\Delta_{i}=0\) (D) None of these

Step-by-Step Solution

Answer

(B) \( \sum_{i=1}^{k} \Delta_{i}=0 \)

Step 1: Understanding the Problem

We need to find the possible third-order determinants from a set of 9 distinct non-zero real numbers and analyze properties about their count and sum.

Step 2: Calculate Number of Determinants

Each third-order determinant is constructed using 3 rows and 3 columns from the 9 numbers, forming a 3x3 matrix. The number of ways to choose 3 numbers out of 9 for each row and column is \( \binom{9}{3} \). For each choice, there are 3! ways to arrange the numbers in each row and the same for each column, so the total number \( k \) is \( 9! / (3!)^2 \).

Step 3: Evaluate the Sum of Determinants

If we choose all possible determinants, for each arrangement of 9 numbers split into a 3x3 matrix, the sum of all possible determinants is zero because the positive and negative permutations will cancel each other out over all combinations.

Step 4: Check for Zero Determinants

A third-order determinant can be zero if any two rows or columns are proportional. Since all numbers are distinct and non-zero, zero determinants can occur depending on the chosen numbers and their resulting arrangement, but it's not guaranteed without specific configuration.

Step 5: Final Selection

Option (B) \( \sum_{i=1}^{k} \Delta_{i}=0 \) states that the sum of all possible third-order determinants equals zero, which aligns with the observations about permutation cancellation.

Key Concepts

Third Order Determinants

A third-order determinant refers to a determinant of a 3x3 matrix. It is calculated by taking three rows and three columns from the total set of numbers, resulting in a determinant value. The formula for a third-order determinant using elements of a 3x3 matrix \[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg),\]where \(A\) = \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\). Calculating third-order determinants requires finding the product of diagonal elements and their subtraction. This algebraic expression determines the unique value called the determinant of the given matrix. To construct a third-order determinant from 9 distinct numbers, it’s about selecting and arranging them within the 3x3 grid for different possible configurations. This exercise involves permutations and combinations. Understanding how to effectively calculate and interpret third-order determinants is crucial for matrix analysis, especially when dealing with advanced mathematical problems such as solving systems of equations.

Combinatorics in Determinants

Combinatorics is the field of mathematics dealing with combinations and permutations. In the context of determinants, it plays a vital role in determining the number of ways to select elements to form a matrix. Here, specifically, from 9 distinct non-zero numbers, we choose groups of 3 for both the rows and columns to form a 3x3 matrix.Each choice involves using combinations denoted by \( \binom{9}{3} \), which calculates how to choose 3 items from a set of 9, regardless of order. In this exercise, once the 3 numbers are selected, they can be arranged within the row or column in factorial \((3!\)) ways.- Calculating for rows: \( \binom{9}{3} \times 3! \).- Similarly, calculate for columns.- Total combination is modified by arranging both, resulting in permutations.This sequential selection of subsets and permutations eventually simplifies the number of configurations to be calculated as per the step-by-step solution, demonstrating the power of combinatorics in calculating determinants across possible matrices.

Properties of Determinants

Determinants have several important properties that are crucial in analysis. Understanding these properties aids in matrix computations and theoretical assessments in linear algebra. - **Linearity**: The determinant is a linear function regarding each row and column. This means that altering matrices by a scalar or adding/subtracting linear combinations from another row or column will affect the determinant predictably. - **Permutation Sign**: The determinant value is affected by the permutations of rows and columns. If you swap any two rows or columns, the sign of the determinant flips. - **Zero Determinant**: If any two rows (or columns) of a matrix are identical or linearly dependent, the determinant is zero. This signifies the lack of uniqueness in solutions of systems or leads to singular matrices, which don't have an inverse. - **Cancellations in Permutations**: As seen in the exercise, the summation of determinants resulting from all permutations of the numbers can cancel out leading to zero, a profound concept related to symmetry and balance in permutations. These properties are not only mathematical rules but are intuitive reflections of underlying geometrical and algebraic structures. Understanding these helps to solve complex mathematical problems precisely.