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

Question

Let \(A\) be an \(n \times n\) matrix such that \(A^{n}=\alpha A\), where \(\alpha\) is a real number different from 1 and \(-1\). Then, the matrix \(A+I_{n}\) is (A) singular (B) non-singular, i.e., invertible (C) scalar matrix (D) None of these

Step-by-Step Solution

Answer

The matrix \(A+I_n\) is non-singular.

Step 1: Understanding the Given Condition

We are given that \(A^{n} = \alpha A\) for an \(n \times n\) matrix \(A\), where \(\alpha\) is a real number different from 1 and -1. This implies that \(A\) satisfies the polynomial \(x^n - \alpha x = 0\).

Step 2: Analyzing the Polynomial Equation

Since \(x(x^{n-1} - \alpha) = 0\), the possible eigenvalues of \(A\) are 0 or the roots of \(x^{n-1} = \alpha\). These possible eigenvalues affect the behavior of \(A + I_n\).

Step 3: Analyzing the Matrix \(A + I_n\)

The eigenvalues of \(A + I_n\) are \(\lambda + 1\), where \(\lambda\) are the eigenvalues of \(A\). If 0 is a root, the corresponding eigenvalue to \(A + I_n\) would be 1. However, if \(A\) has no zero eigenvalue, \(A + I_n\) is invertible.

Step 4: Considering the Non-zero Eigenvalues

As from the polynomial \(x^n - \alpha x = 0\), if \(\alpha = 0\), then \(A\) has an eigenvalue 0 (not possible since \(\alpha eq 0\)). If the eigenvalues are non-zero, adding identity never makes \(A + I_n\) singular.

Step 5: Conclusion Based on Eigenvalues

Since \(\alpha eq 0\), and \(\alpha eq 1\) or \(-1\), the eigenvalues of \(A\) (that would make \(A + I_n\) singular) are not allowed, confirming \(A + I_n\) is non-singular.

Key Concepts

Eigenvalues of Matrices

Eigenvalues are crucial to understanding many properties of matrices. They are the scalars, \( \lambda \), associated with a matrix \( A \) that satisfy the equation \( A\mathbf{v} = \lambda\mathbf{v} \) for some non-zero vector \( \mathbf{v} \). These vectors, \( \mathbf{v} \), are known as eigenvectors.
Eigenvalues can provide insights into the matrix behavior, such as stability and response to linear transformations.

• Zero Eigenvalues: If a matrix has zero as an eigenvalue, it indicates that the matrix is singular (non-invertible).
• Non-zero Eigenvalues: Non-zero eigenvalues suggest that the matrix could be invertible, depending on the specific nature of these eigenvalues.

In the context of the problem, the eigenvalues of matrix \( A \) help determine whether \( A + I_n \) is invertible. Since eigenvalues of \( A + I_n \) are derived from those of \( A \) by adding 1, ensuring these values are non-zero helps in confirming the invertibility of \( A + I_n \).

Polynomial Equations in Linear Algebra

Polynomial equations play a central role in determining the characteristics and behaviors of matrices.
In the given problem, the polynomial equation \( x^n - \alpha x = 0 \) arises from the matrix equation \( A^n = \alpha A \), and serves to identify potential eigenvalues of the matrix \( A \).

• The roots derived from the polynomial, such as 0 or those satisfying \( x^{n-1} = \alpha \), are candidate eigenvalues.

Understanding these roots is pivotal as they determine matrix characteristics such as invertibility and rank.
Additionally, the polynomial nature allows us to explore higher order behaviors of matrices, uncovering deeper insights into their long-term transformations.

Matrix Theory

Matrix theory is a branch of mathematics focusing on the study of arrays of numbers and their transformations.
The exercise in question taps into several key concepts within this theory:

• Invertibility of Matrices: A matrix is invertible if there exists another matrix that when multiplied with yields the identity matrix.
• Identity Matrix: The matrix \( I_n \) plays an important role in ensuring changes in eigenvalues, as seen in the sum \( A + I_n \).
• Singular vs Non-Singular: Matrices are singular when they lack an inverse. The eigenvalues guide this distinction.”

Studying how matrices behave when transformed or as part of equations like \( A^n = \alpha A \), is key to traversing matrix theory floors, particularly in understanding more complex interactions within and between vectors.