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

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 these

Step-by-Step Solution

Answer

The answer is option (A): \((x-1)^{n-1}(x+n-1)\).

Step 1: Understanding the Matrix

The given determinant is of an \(n \times n\) matrix, where all the diagonal elements are \(x\) and all the off-diagonal elements are \(1\). This is a well-known type of matrix known as a circulant matrix, where we observe a unique pattern in its structure.

Step 2: Characteristic Polynomial

For matrices like this, we often look at their characteristic polynomial. A common result of circulant matrices with this structure is that their determinant is \((x-1)^{n-1}(x+n-1)\). This results from considering how the constant off-diagonal values affect the determinant's value.

Step 3: Confirming the Formula

We can confirm the result by using properties of determinants, specifically leveraging the fact that the matrix can be viewed as a perturbation of the identity matrix by a rank-1 matrix. Calculating directly, we reduce the problem to a simpler polynomial which matches the expected result: \((x-1)^{n-1}(x+n-1)\).

Step 4: Match with Options

Review the given options, noting that the determined value \((x-1)^{n-1}(x+n-1)\) corresponds to option \(A\). This verifies our solution using mathematical properties and comparing it to known results.

Key Concepts

Circulant Matrix

In linear algebra, a circulant matrix is a special type of matrix where each row vector is a cyclic shift of the previous one. In other words, each row is obtained by moving the last element of the previous row to the first position and shifting all other elements one position to the right. This repetitive pattern simplifies many computations and offers symmetrical properties that are useful in determinant calculations.
For example, consider a simple 3x3 circulant matrix:

• First row: \[\begin{bmatrix} a & b & c \end{bmatrix}\]
• Second row: \[\begin{bmatrix} c & a & b \end{bmatrix} \]
• Third row: \[\begin{bmatrix} b & c & a \end{bmatrix} \]

Circulant matrices are useful in many fields such as signal processing and discrete Fourier transforms because they are diagonalizable by the Fourier matrix, making calculations computationally efficient.

Characteristic Polynomial

The characteristic polynomial of a matrix is a polynomial expression which is crucial in determining the eigenvalues of the matrix. For a matrix \(A\), the characteristic polynomial is defined by \( \det(A - \lambda I) \), where \( \lambda \) represents an eigenvalue and \( I \) is the identity matrix of the same order as \( A \).
The roots of the characteristic polynomial give us the eigenvalues which are significant in understanding matrix behavior, including stability and resonance properties in different applications. For circulant matrices, the characteristic polynomial has a particularly structured form due to the uniform shifting pattern of its rows.
These properties allow the determinant of the matrix to be more easily calculated, as was demonstrated in the original exercise solution. The knowledge of the characteristic polynomial helps confirm the formula: \((x-1)^{n-1}(x+n-1)\). Recognizing this polynomial provides a shortcut to understanding the determinant of our original circulant matrix.

Matrix Diagonal

The diagonal of a matrix consists of the elements that stretch from the top left to the bottom right corner. In mathematical terms, it is formed by elements \(a_{ii}\) for all \(i\). Understanding the properties of a matrix's diagonal is vital because it often dictates key attributes of the matrix, especially in terms of determinant calculation.
In the context of the exercise, the matrix diagonal is filled with the element \(x\), while the off-diagonal elements are \(1\). This configuration classifies it as a type of circulant matrix, which significantly impacts its determinant properties. For matrices that primarily consist of repeated elements along the diagonal or off-diagonal, modified determinant rules or simplifications often apply.
Recognizing and leveraging the uniform pattern in such matrices can aid in predicting their determinant behavior, as observed in the exercise's comprehensive solution.

Determinant Properties

The determinant is a scalar attribute of a square matrix that provides a significant amount of information about the matrix itself. It can indicate the volume distortion by the matrix in geometric transformations and is essential for solving systems of linear equations. Key properties include:

• If the determinant is zero, the matrix is singular, indicating that it does not have an inverse.
• Determinants of triangular matrices (upper or lower) are the product of the diagonal elements.
• A matrix and its transpose have the same determinant.
• The addition of a multiple of one row to another row does not change the determinant.

For the specific matrix in the exercise, determinant properties such as linearity and column-row operations were used to simplify calculus, resulting in the formula \((x-1)^{n-1}(x+n-1)\) applicable to circulant matrices with similar configurations. Understanding these properties allows for efficient computation of determinants in more complex matrices.