Problem 99 - 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 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

Correct statements: B and C

Step 1: Analyzing Statement A

Consider two matrices, \(A\) and \(B\), both \(n\times n\). \(A^{-1}\) is the inverse of \(A\), so \(A^{-1}B\) and \(BA^{-1}\) are both \(n\times n\) matrices. However, there is no general property that implies \(A^{-1}B\) and \(BA^{-1}\) have the same eigenvalues. Therefore, this statement is not necessarily true.

Step 2: Analyzing Statement B

Consider matrices \(A\) and \(P\) of the same order with \(P\) being non-singular. The transformation \(P^{-1}AP\) is called a similarity transformation. It is a known property that similar matrices have the same eigenvalues. Therefore, \(A\) and \(P^{-1}AP\) have the same characteristic roots, making this statement true.

Step 3: Analyzing Statement C

For square matrices \(A\) and \(B\) of the same order, if \(AB\) and \(BA\) are both defined, then they have the same eigenvalues. This is a known result in linear algebra where the product of eigenvalues of \(AB\) and \(BA\) is the same, even though \(AB\) and \(BA\) do not necessarily have the same matrix structure. Hence, this statement is correct.

Step 4: Conclusion

Based on the analysis, Statements B and C are correct while Statement A is not. Therefore, the correct answer is not all of the given statements; it specifically excludes (A). Thus, only Statements B and C are true for the eigenvalue properties described.

Key Concepts

Characteristic Polynomial

The characteristic polynomial of a matrix is a fundamental concept in linear algebra. When you have an \(n\times n\) matrix \ A \, you modify it by subtracting \lambda I\, where \lambda\ is a scalar, and \I\ is the identity matrix of the same size. The resulting matrix \(A- lambda I\) is known as the characteristic matrix.
To find the characteristic polynomial, you calculate the determinant of this characteristic matrix, \|A- lambda I|\. This determinant gives you a polynomial of degree \(n\) in \lambda\, known as the characteristic polynomial. The roots of this polynomial, i.e., the values of \lambda\ for which \|A- lambda I|=0\, are called the eigenvalues of the matrix.

• The characteristic polynomial provides critical information about the matrix's behavior and properties.
• The degree of the characteristic polynomial matches the number of eigenvalues, counted with their multiplicities.
• This polynomial helps predict stability in systems of differential equations and in systems theory, among other applications.

Determinant of a Matrix

The determinant of a matrix is a scalar value that can provide invaluable insights into the properties of the matrix. For an \(n\times n\) matrix \(A\), the determinant \|A|\ is a single number.
This number tells us several important things:

• If the determinant is zero, the matrix is singular, meaning it does not have an inverse.
• If it's non-zero, the matrix is non-singular and has an inverse.

The determinant plays a crucial role in calculating the characteristic polynomial. Specifically, the determinant of \(A- lambda I\) leads us to the characteristic polynomial.
In terms of eigenvalues, it is noteworthy that the product of a matrix's eigenvalues is equal to its determinant. This property can simplify calculations tremendously, especially when dealing with larger matrices, providing a direct way to check for singularity or to find numerical solutions of matrix equations.

Matrix Similarity

Matrix similarity is a concept indicating that two matrices can represent the same linear transformation under different bases. If matrices \(A\) and \(B\) are of the same order, and there exists an invertible matrix \(P\) such that \(B = P^{-1} A P\), then \(A\) and \(B\) are said to be similar.
An essential property to remember about similar matrices is that they always share the same eigenvalues. This is because the characteristic polynomial is invariant under similarity transformations:

• This means that even though the actual matrix representation (i.e., the form of the matrix) might change, the eigenvalues -- which are fundamentally related to the matrix's structure -- remain the same.
• The similarity transformation can simplify complex matrix computations by transforming a matrix to a more convenient form without altering its spectrum (set of eigenvalues).

Understanding matrix similarity is vital for broader insights into matrix theory and applications in various fields, including system dynamics and quantum mechanics.

Non-Singular Matrix

A non-singular matrix, often called an invertible matrix, is one that has an inverse. This means there exists another matrix that, when multiplied with the given matrix, yields the identity matrix.
Determining whether a matrix is non-singular is straightforward: a matrix is non-singular if its determinant is non-zero. This property makes non-singular matrices extremely useful:

• They allow solving systems of linear equations using matrix operations.
• In various computations, the availability of an inverse simplifies solving matrix equations.

Non-singular matrices are crucial in transformations and optimizations, as they guarantee unique solutions and ensure that transformations are reversible. Recognizing a non-singular matrix and understanding its properties allow for exploiting its advantages in practical applications ranging from statistics to engineering.