Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant of \(n\)th order, being given by \(\left|\begin{array}{cccc}x & 1 & 1 & \ldots \\ 1 & x & 1 & \ldots \\ 1 & 1 & x & \ldots \\ \ldots & \ldots & \ldots & \ldots\end{array}\right|\), is (A) \((x-1)^{n-1}(x+n-1)\) (B) \((x-1)^{n}(x+n-1)\) (C) \((1-x)^{n-1}(x+n-1)\) (D) None of thes

Step-by-Step Solution

Answer

The value of the determinant is option (A): \((x-1)^{n-1}(x+n-1)\).

Step 1: Understand the determinant format

We have a determinant of order \(n\) where the diagonal elements are \(x\) while all off-diagonal elements are 1. This is a special type of determinant called a circulant matrix.

Step 2: Rewrite the determinant

Represent the determinant as \( D_n = \begin{vmatrix} x & 1 & 1 & \cdots & 1 \ 1 & x & 1 & \cdots & 1 \ 1 & 1 & x & \cdots & 1 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 1 & 1 & 1 & \cdots & x \end{vmatrix} \). This pattern continues through all rows and columns.

Step 3: Calculate determinant for small values of n

Calculate for small values like \( n = 2 \): \[ D_2 = \begin{vmatrix} x & 1 \ 1 & x \end{vmatrix} = x^2 - 1 \]. For \( n = 3 \): \[ D_3 = \begin{vmatrix} x & 1 & 1 \ 1 & x & 1 \ 1 & 1 & x \end{vmatrix} = x^3 - 3x + 2 \].

Step 4: Use the formula for determinant of such matrices

If a matrix has \(x\) on the diagonal and 1s elsewhere, the determinant is given by \((x-1)^{n-1}(x+n-1)\). Verify this with values calculated in Step 3.

Step 5: Match with given options

From our derived formula \((x-1)^{n-1}(x+n-1)\), it matches with option (A).

Key Concepts

Circulant Matrix

Circulant matrices are fascinating structures in matrix algebra, characterized by their unique repetitive patterns. They belong to a special class of matrices where each row is a cyclic permutation of the row above it. This means that if you take any row in a circulant matrix and move its first element to the end of the row, you'll get the next row. This pattern continues throughout the entire matrix, creating a circular symmetry.

In the context of determinants, circulant matrices often reveal intriguing properties. For example, when calculating the determinant of a circulant matrix where diagonal elements are constant, different techniques can be employed that simplify the computation, due to their repetitive nature. This makes solving problems like the given exercise more manageable and provides insight into how these matrices work. Understanding circulant matrices helps improve efficiency in matrix computations, as their patterns can reduce complexity in mathematical operations.

Matrix Algebra

Matrix algebra is a significant branch of mathematics dealing with matrices and their properties. It involves a variety of operations including addition, subtraction, multiplication, and finding determinants. These operations are foundational in solving systems of equations, transformations in geometry, and many other areas of applied mathematics.

Determinants are an essential concept in matrix algebra. They are scalar values that can be calculated from square matrices, providing valuable information such as whether a system of linear equations has a unique solution. In matrix algebra, understanding how to compute determinants, especially for complex matrices like circulant ones, is crucial.

• Matrix addition involves combining two matrices by adding their respective elements.
• Matrix multiplication involves more complex operations, essential for transformations and systems of equations.
• Determinants provide insight into the properties of matrices, influencing systems of equations and matrix inverses.
  

Mastering these operations allows students to tackle increasingly complex mathematical and real-world problems.

Order of Determinants

The order of a determinant refers to the number of rows (or columns) in a square matrix. It is denoted as the matrix's size, such as 2x2, 3x3, or in general, nxn. The order of the determinant dictates the complexity of calculating it; larger orders generally mean more complex calculations.

For small matrices, such as 2x2 or 3x3, the formulae are fairly simple. A 2x2 determinant can be calculated using the formula \( ad - bc \), where \( a, b, c, \) and \( d \) are elements of the matrix. As the order increases, more elaborate methods are often needed. However, understanding the principles of smaller matrices provides a foundation for tackling larger ones.

• The order determines the dimensions and complexity of a matrix.
• Larger orders require systematic approaches or formulae to simplify calculation.
• Each increase in order adds layers to the computation, demanding clear understanding of underlying principles.
  

Grasping the concept of the order of determinants is vital for predicting how modifications to a matrix will alter the determinant, impacting the matrix's behavior and properties.