Problem 99 - A Complete Resource Book in Mathematics for JEE Main

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 correct? (A) If \(A, B\) are \(n\) rowed square matrices and \(A\) is non-singular, then \(A^{-1} B\) and \(B A^{-1}\) has same character-istic roots. (B) If \(A\) and \(P\) are square matrices of same order and \(P\) is non-singular, then \(A\) and \(P^{-1} A P\) have same characteristic roots. (C) If \(A\) and \(B\) be two square matrices of same order, then \(A B\) and \(B A\) have same characteristic roots. (D) All of these

Step-by-Step Solution

Answer

All statements (A), (B), and (C) are correct.

Step 1: Evaluate Statement (A)

We need to determine if the matrices \(A^{-1}B\) and \(BA^{-1}\) have the same characteristic roots. Both matrices \(A^{-1}B\) and \(BA^{-1}\) are similar to the same matrix \(A^{-1}BA^{-1}\), thus they share the same characteristic roots. Therefore, statement (A) is correct.

Step 2: Evaluate Statement (B)

This statement says that matrices \(A\) and \(P^{-1}AP\) have the same characteristic roots if \(P\) is non-singular. Two similar matrices, such as \(A\) and \(P^{-1}AP\), indeed share the same characteristic polynomial and roots. Thus, statement (B) is correct.

Step 3: Evaluate Statement (C)

The statement claims that the products \(AB\) and \(BA\), where both are square matrices of the same size, have the same characteristic roots. This is true, as the eigenvalues of products \(AB\) and \(BA\) are indeed the same. Therefore, statement (C) is correct.

Step 4: Conclude Result

Since all the evaluations have shown that each statement is correct, all the given statements (A), (B), and (C) are true.

Key Concepts

Eigenvalues

Eigenvalues are fundamental in linear algebra, particularly in the study of matrices. Imagine a matrix as a transformation in a space. Eigenvalues are the factors by which the transformation scales the eigenvectors.

• To find eigenvalues, you must solve the characteristic equation \(|A - \lambda I| = 0\), where \( A \) is the matrix and \( I \) is the identity matrix.
• This equation is derived by seeking values of \(\lambda\) such that \( (A - \lambda I)X = 0\).

Eigenvalues can tell us a lot about the matrix, such as whether it's singular (i.e., one of its eigenvalues is zero), and they play a crucial role in determining the invertibility of the matrix. When used in various contexts, eigenvalues are crucial for understanding stability, vibration modes, and the behavior of systems over time in applied mathematics.

Matrix Similarity

Matrix similarity is a fascinating concept in linear algebra that shows a deeper connection between different matrices. Two matrices, say \( A \) and \( B \), are similar if there exists a non-singular matrix \( P \) such that \( B = P^{-1}AP \).

• Similar matrices have identical eigenvalues and characteristic polynomials, which makes them very useful in simplifying problems.
• Importantly, similar matrices represent the same linear transformation, but in different bases of the vector space.

This is why statement (B) from the exercise is correct: when you change the basis of the matrix with a non-singular matrix \( P \), the eigenvalues remain unchanged. This reveals the intrinsic properties of the linear transformation, unaltered by the choice of basis.

Determinant

The determinant of a matrix is a scalar that encapsulates many characteristics of the matrix itself. For an \( n \times n \) matrix \( A \), the determinant is denoted by \( |A| \) and can be computed using various methods, such as expansion by minors or row reduction.

• The determinant offers insights into properties such as whether the matrix is invertible; a non-zero determinant indicates an invertible matrix.
• Interestingly, the determinant is also the product of a matrix's eigenvalues, highlighting a deep relationship between these concepts.

When it comes to understanding transformations, the determinant helps determine whether areas or volumes are stretched or shrunk. Hence, it has broad applications in geometry, physics, and engineering, providing a numerical summary of a transformation's network of effects on space.