Problem 98 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(A=\left[a_{i j}\right]\) be an \(n \times n\) matrix. The matrix \(A-\lambda I\) is called the characteristics matrix of \(A\), where \(\lambda\) is a scalar and \(I\) is the identity matrix. The determinant \(|A-\lambda I|\) is a non-null polynomial of degree \(n\) in \(\lambda\) and is called the characteristic polynomial of \(A\). The equation \(|A-\lambda I|=0\) is called the characteristic equation of \(A\) and its roots are called the characteristic roots or latent roots or eigen values of \(A\). The set of all eigenvalues of the matrix \(A\) is called the spectrum of \(A\). The product of the eigenvalues of a matrix \(A\) is equal to the determinant \(A\). Which of the following statements are true? If \(A\) is any \(n \times n\) matrix and \(\lambda\) is a characteristic root of \(A\), then (A) \(A\) and \(A^{\prime}\) have the same characteristic roots (B) \(k \lambda\) is a characteristic root of \(k A\) ( \(k\) being scalar) (C) \(\lambda^{n}\) is a characteristic root of \(A^{n}\) ( \(n\) being positive integer) (D) \(\frac{1}{\lambda}\) is a characteristic root of \(A^{-1}\)

Step-by-Step Solution

Answer

Statements (A), (C), and (D) are true.

Step 1: Understand the Characteristic Roots

The characteristic roots or eigenvalues of a matrix are the solutions to the characteristic equation \(|A - \lambda I| = 0\). These roots are inherent properties of the matrix and hold significance in various applications, such as diagonalization.

Step 2: Check Statement (A)

Both matrix \(A\) and its transpose \(A'\) have the same characteristic polynomial, and consequently the same eigenvalues, because determinants and hence characteristic polynomials are invariant under transposition.

Step 3: Check Statement (B)

If \(kA\) is a matrix multiplied by a scalar \(k\), its characteristic polynomial is \(|kA - \lambda I|\). This is equivalent to \(k^n|A - (\lambda/k)I|\). This implies that the eigenvalues of \(kA\) are \(k\lambda_1, k\lambda_2, \ldots, k\lambda_n\) where \(\lambda_1, \lambda_2, \ldots, \lambda_n\) are the eigenvalues of \(A\), hence the statement is not correctly reflecting the dependency on \(k\).

Step 4: Check Statement (C)

The eigenvalues of \(A^n\) are \(\lambda_1^n, \lambda_2^n, \ldots, \lambda_n^n\) if \(\lambda_1, \lambda_2, \ldots, \lambda_n\) are the eigenvalues of \(A\). Therefore, the statement that \(\lambda^n\) is an eigenvalue of \(A^n\) is correct.

Step 5: Check Statement (D)

If \(\lambda\) is a characteristic root of \(A\), then \(Ax = \lambda x\) for some non-zero vector \(x\). For \(A^{-1}\), the equation \(A^{-1}y = \frac{1}{\lambda} y\) holds, implying that \(\frac{1}{\lambda}\) is indeed an eigenvalue of \(A^{-1}\).

Key Concepts

Characteristic matrix

A characteristic matrix is an important concept in linear algebra, especially when studying the eigenvalues of a matrix. It's formed by subtracting a scalar multiple of the identity matrix from the matrix in question.
For a given matrix \( A \) and a scalar \( \lambda \), the characteristic matrix becomes \( A - \lambda I \), where \( I \) is the identity matrix of the same size as \( A \).

• The identity matrix, \( I \), is crucial because it allows only the scalar \( \lambda \) to be subtracted from the diagonal elements of \( A \), leaving other elements unchanged.
• This structure facilitates the study of eigenvalues as the determinant of this matrix results in the characteristic polynomial.

Understanding and calculating the characteristic matrix is the first step in finding eigenvalues, which are the solutions to the characteristic equation obtained from it, \(|A - \lambda I| = 0\).

Characteristic polynomial

The characteristic polynomial of a matrix unveils a lot about its structural properties.
Derived from the characteristic matrix, the characteristic polynomial is found by taking the determinant \(|A - \lambda I|\), yielding a polynomial of degree \( n \) for an \( n \times n \) matrix.

• This polynomial has the form \( a_{n}\lambda^n + a_{n-1}\lambda^{n-1} + \ldots + a_0 \) where \( a_i \) are coefficients determined by the elements of the original matrix \( A \).
• The roots of this polynomial are the eigenvalues of the matrix, revealing important insights such as stability and dynamic response in systems theory.

Finding the characteristic polynomial is a key step toward determining these eigenvalues and fully understanding the behavior of matrix-related systems.

Transpose of a matrix

Transposing a matrix involves switching its rows with its columns, which seems simple but has deep implications for its properties.
Notably, both a matrix \( A \) and its transpose \( A^T \) share the same eigenvalues.

• This invariance occurs because the determinant of a matrix remains unchanged upon transposition, which implies that the characteristic polynomials (and thus the roots, which are the eigenvalues) are identical.
• This property helps in simplifying problems and understanding deeper matrix behaviors, such as symmetry influences and simplifying computations.

Transposing retains certain properties, making it a powerful tool in linear transformations and facilitating the analysis of matrices via their eigenvalues.

Matrix inverse

The inverse of a matrix, denoted \( A^{-1} \), plays a pivotal role in solving linear equations and understanding transformations.
Of special interest is how the eigenvalues of \( A \) relate to those of \( A^{-1} \).

• If \( \lambda \) is an eigenvalue of \( A \), then \( \frac{1}{\lambda} \) becomes an eigenvalue of \( A^{-1} \).
• This relationship is fundamental when exploring properties of systems, as it helps to understand the behavior of systems under inversions or transformations.

Knowing about matrix inverses and their connection to eigenvalues allows for deeper insight into systems' stability, control, and response characteristics in applied contexts.