Problem 35 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The value of 'n' is any integer, option (D).

Step 1: Recognize the Equation

We are given the sum of two determinants equal to zero: \[\left|\begin{array}{ccc} a & a+1 & a-1 \ -b & b+1 & b-1 \ c & c-1 & c+1 \end{array}\right|+ \left|\begin{array}{rrr} a+1 & b+1 & c-1 \ a-1 & b-1 & c+1 \ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^{n} c \end{array}\right| = 0.\]

Step 2: Evaluate First Determinant using Properties

To evaluate the first determinant, apply elementary row operations such as adding, subtracting, or rearranging rows. Consider simplifications like row exchanges or factorization if applicable.

Step 3: Examine Special Symmetries or Patterns

Consider any noticeable patterns or symmetries in the arrangement of terms and evaluate the role of each element in contributing to the determinant value.

Step 4: Evaluate Second Determinant Based on Powers of -1

The second determinant involves powers of (-1) terms. Explicitly substitute these terms and factor in the impact of any specific values of \(n\) (particularly focus on even vs. odd \(n\)) to reveal if these terms simplify to zero.

Step 5: Interpretation and Solution

If terms simplify to zero when grouped by particular values of \(n\), deduce that both components sum correctly for specific \(n\). Analyze patterns to reach a logical conclusion.

Step 6: Combine Results

Using insights from earlier steps, conclude the solution about which values of \(n\) satisfy the equation. Here, odd powers of (-1) terms simplify under integer assumptions, leading us mostly towards symmetry or dependent primarily on even or odd decisions.

Key Concepts

Elementary Row Operations

Elementary row operations are a powerful tool used often to simplify matrices and calculate determinants. These operations include row swapping, row multiplication, and adding or subtracting the multiples of one row from another. Understanding how and when to apply these operations can significantly ease calculations by promoting simplifications.

Applying these operations to evaluate determinants is especially useful if you can set rows to zeros or create simpler patterns. For instance:

• Row Swapping: Switching any two rows changes the sign of the determinant.
• Row Multiplication: Multiplying a row by a number scales the determinant by the same factor.
• Row Addition/Subtraction: Adding or subtracting a multiple of one row to another doesn’t change the determinant's value.

These operations help reduce complex matrices and make the computation of determinants more manageable. In our problem, by applying these operations and looking for simplifications, you can better analyze each determinant's properties.

Symmetry in Determinants

Symmetry in determinants refers to recognizable patterns which often simplify computations. These patterns may result in some terms canceling out or repeating in a symmetry, simplifying the calculation process.

For the given problem, examining symmetries involves:

• Identifying symmetric arrangement of elements that might lead to cancellations.
• Understanding how symmetric matrix properties can lead to zero-valued determinants under particular transformations or operations.

Noticing such symmetrical structures allows you to anticipate more manageable determinant values or even pinpoint when certain configurations may lead to zero. This perspective can be particularly useful when simplifying the determinants through elementary row operations, as evidenced by the problem's need to recognize special patterns.

Powers of Negative One in Determinants

The powers of negative one, or \((-1)^n\), play an intriguing role in determinant calculation. They introduce a periodic behavior depending on whether the exponent \(n\) is even or odd. This behavior can have significant implications for the matrix's resulting value.

In the context of this exercise:

• A power of \( (-1)^n \) with an even \(n\) yields a positive one, while an odd \(n\) yields a negative one.
• This alteration can profoundly impact whether the determinant sum refines to zero.

Recognizing these powers helps when evaluating the determinant that involves exponential expressions, as they directly impact the resultant positivity or negativity of rows or columns.

By calculating determinants under the correct interpretation of \((-1)^n\), you can more accurately determine which configurations of \(n\) hold under the equation's constraint. This insight is pivotal for arriving at an accurate solution, as illustrated by distinguishing between odd or even integer assumptions in powers affecting the determinant outcome directly.