Problem 35 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(a, b, c\) be such that \(b(a+c) \neq 0\). If \(\left|\begin{array}{ccc}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|\) \(+\left|\begin{array}{rrr}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^{n} c\end{array}\right|=0\), then the value of ' \(n\) 'is (A) zero (B) any even integer (C) any odd integer (D) any integer

Step-by-Step Solution

Answer

D: any integer

Step 1: Understand the Problem

We are given two determinants and the condition that their sum equals zero. We need to find the value of \( n \) that satisfies this condition. The determinants' sizes are 3x3, and they both involve the variables \( a \), \( b \), and \( c \).

Step 2: Simplify the Determinant Expression

Each determinant is a 3x3 matrix. The sum of these two determinants is given to be zero. Therefore, set up the equation:\[\left|\begin{array}{ccc}a & a+1 & a-1 \ -b & b+1 & b-1 \ c & c-1 & c+1\end{array}\right| + \left|\begin{array}{rrr}a+1 & b+1 & c-1 \ a-1 & b-1 & c+1 \ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^{n} c\end{array}\right| = 0\]

Step 3: Expand the Determinants

Calculate each determinant separately. For a 3x3 determinant, the value is given by adding and subtracting the cross-products of each element of the first row and the corresponding 2x2 minor. Calculate:1. For the first determinant:\[ abc - ab(a-1) - ac(a+1) - bc(b-1) + c(ab) + ba(c-1) \]2. For the second determinant:\[ ((-1)^{n+2} a)((b-1)(c+1) - (b+1)(c-1)) + ((-1)^{n+1} b)((a-1)(c-1) - (a+1)(c+1)) + ((-1)^n c)((a+1)(b+1) - (a-1)(b-1)) \]

Step 4: Solve for n

For the equation to hold true (equal zero), the terms from each determinant should cancel each other out. Notice each entry's presence in both determinants suggests cancellation. After simplification, equate both the determinants:\[\begin{aligned} & abc - ab(a-1) - ac(a+1) - bc(b-1) + c(ab) + ba(c-1) \ &= a(c-b) - b(a-c) \pm 2ab \& = 0 \end{aligned}\]The simplification reveals that the equation can only be consistent for specific arrangements and only occurs if the effects of \( n \) alternate between the two determinants. Assess whether \( n \) can be any integer based on the symmetry: \( n \) influences sign changes through powers of -1.

Step 5: Choose the Correct Answer

From Step 4, it is evident that the alternating signs due to \( n \) can lead the expressions to balance in the presence of the alternating power, allowing any value of \( n \). Therefore, the answer that satisfies this condition is (D): any integer.

Key Concepts

Matrix Algebra

Matrix algebra forms the backbone for understanding and solving problems involving determinants. At its core, matrix algebra deals with the manipulation of matrices, which are grid-like arrays of numbers, variables, or expressions, arranged in rows and columns. When performing operations on matrices, such as addition, subtraction, or finding the determinant, specific rules and properties apply. These enable us to derive meaningful results, just as in our exercise.

Determinants, a key operation within matrix algebra, are scalar values that can be computed from a square matrix. In the context of this problem, we're working with the 3x3 matrices. Calculating determinants involves finding the sum of specific products subtracted from one another, giving us insights into the properties of the matrix like invertibility or the volume spanned by its columns. Mastering these basics is crucial for effectively tackling exercises like this one that involve matrix determinants.

3x3 Matrix

A 3x3 matrix is a matrix that consists of three rows and three columns, making it a 3x3 grid of elements. Each element in a 3x3 matrix can be a number or a variable. In the exercise, we encounter two 3x3 matrices, each contributing to the computation of their respective determinants.

To compute a 3x3 matrix determinant, we use the rule of Sarrus or expansion by minors. This typically involves taking each element in the first row and multiplying it by the determinant of the 2x2 matrix that remains after removing the element's row and column. The result involves a combination of positive and negative terms, reflecting which subspaces affect the whole system.

• The first element helps establish the scale of influence across the grid.
• The other elements follow a similar principle but account for larger intertwined influences.

Understanding how a 3x3 matrix behaves prepares us for more complex scenarios involving variables and algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operators. In the context of our matrix exercise, variables like \(a\), \(b\), and \(c\) make the matrix elements. Handling these algebraic expressions involves simplifying them during the process of computing the determinant.

When expanding the determinants, algebraic expressions require careful manipulation:

• Combine like terms, which helps reduce complexity.
• Use identity properties to simplify further, especially when similar terms appear.
• Account for positivity or negativity due to powers of \(-1\), which affects final outcomes in matrices like in our determinant setup.

Breaking these parts down aids in finding intuitive patterns and ensuring an accurate solution that balances both algebraic structures and numeric values within the matrix systems.

Integer Properties

Integer properties come into play in this exercise, particularly through the powers of \(-1\) associated with \(n\) in some matrix elements. Integers like \(n\) do not just alter the sign of terms, but also dictate how expressions combine or cancel out in competing matrices.

Core integer properties reinforced here include:

• The ability of integers to pivot the influence of positive and negative signs.
• Understanding that even integers produce positive signs, while odd integers flip signs.

Recognizing these properties leads to determining the value of \(n\). The exercise concluded that the flexibility of \(n\) allows for a value that can oscillate any integer without disrupting the determinant's balance, confirming that any integer is consistent.