Problem 15 - A Complete Resource Book in Mathematics for JEE Main

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 nonzero 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

The correct answers are (B) and (C).

Step 1: Understanding Determinants

A third-order determinant involves a 3x3 matrix. From 9 unique numbers, a single 3x3 matrix can be created by selecting any 3 numbers for each row of the matrix.

Step 2: Calculating the number of determinants

To calculate the number of possible third-order determinants, determine how many 3x3 matrices can be made from 9 unique numbers. Choose 3 numbers for the first row (9 options), 3 for the second row (6 options), and 3 for the third row (3 options). This means the total number of possible determinants is the number of ways to arrange 9 numbers in groups of 3, i.e., \(\frac{9!}{(3!)^3}\).

Step 3: Evaluating Determinants

The sum of all possible determinants is zero because for every arrangement of a 3x3 matrix that gives a determinant \(\Delta\), there exists another arrangement where the rows are permuted or the columns are permuted (generating an opposite sign), thereby cancelling out each other when summed.

Step 4: Exploring Non-zero Determinants

All entries of the matrix being non-zero and distinct does not guarantee any determinant is zero, but with permutations and possible arrangements, a few could turn out zero because of geometric properties or symmetric configurations.

Key Concepts

Combinatorics in Determinants

In the context of determinants, combinatorics plays a crucial role when determining the number of possible third-order determinants that can be formed from a set of distinct elements. Combinatorics involves understanding how these elements can be selected and arranged to form matrices. In particular, a third-order determinant necessitates designing 3x3 matrices, which are constructed by picking any three elements for each row.

Consequently, the combinatorial aspect is crucial in calculating the possible combinations. Here, we use the formula for combinations to determine the arrangement of 9 unique numbers into a 3x3 matrix. This requires choosing 3 numbers for the first row, another 3 for the second row, and 3 for the third row. The total number of third-order determinants equates to \(\frac{9!}{(3!)^3}\). This formula accounts for the permutation of numbers across rows while considering each row's arrangement. Hence, combinatorics reveals all plausible configurations of determinants using the distinct integers.

Properties of Determinants

Determinants exhibit a set of unique properties that aid in comprehending their mathematical behavior. One of the essential properties is linearity, which states that the determinant of a matrix will change linearly if one row (or column) is altered. Furthermore, determinants have a multiplicative property, which implies that the determinant of a product of matrices equals the product of their determinants.

A third-order determinant is denoted mathematically by det(A) for a matrix A. One intriguing attribute of determinants involves the cancellation properties. For every matrix arrangement generating a determinant \( \Delta \), there's a corresponding matrix with permuted rows or columns producing \(-\Delta\). Hence, when all these determinants are summed, the result is zero. This fascinating characteristic underlines the balancing nature of determinant sums.

Matrix Arrangements

Matrix arrangements refer to how elements within a matrix are set into rows and columns. These arrangements are significant when considering determinants, as they influence the overall value. For a 3x3 matrix, its arrangement demands selecting 3 numbers for each of the three rows from a larger set of numbers. This step results in diverse matrix configurations, contingent on the chosen numbers and their order.

The manner in which numbers are arranged determines the determinant's sign and magnitude. Arrangement permutations, such as swapping two rows, lead to changes in the determinant's value and sign. This is central to understanding determinants because an arrangement reflects not merely a collection of numbers but a more structured product mirroring properties like symmetry or skew-symmetry.

Permutations in Matrices

Permutations within matrices are vital for understanding determinants, as they help to establish different matrix forms by reallocating numbers. A permutation alters the order of entries, and an essential aspect of determinants is how these permutations affect their values. When elements or rows are permuted in a matrix, the determinant might change its sign, representing an essential feature in linear algebra.

The permutation aspect is clear when considering operations like row swaps. Each row interchange within a matrix transposes a positive determinant into its negative counterpart and vice versa, showcased by the equation \( det(A') = -det(A) \) if A' is obtained by swapping two rows of matrix A. Understanding these permutations helps in realizing how determinants are not static but are influenced by geometrical and algebraic manipulations.