Problem 97 - 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 \(A\). The given values of the matrix \(A=\left[\begin{array}{rrr}1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4\end{array}\right]\) are (A) \(4,-2,-2\), (B) \(-4,2,-2\) (C) \(-4,2,2\) (D) \(4,-4,2\)

Step-by-Step Solution

Answer

The eigenvalues of matrix A are (A) 4, -2, -2.

Step 1: Determine A - λI

To solve for the characteristic polynomial, we must compute the matrix \( A - \lambda I \), where \( I \) is the identity matrix. Given \( A = \begin{bmatrix} 1 & -3 & 3 \ 3 & -5 & 3 \ 6 & -6 & 4 \end{bmatrix} \), the identity matrix \( I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). Thus, \( A - \lambda I = \begin{bmatrix} 1-\lambda & -3 & 3 \ 3 & -5-\lambda & 3 \ 6 & -6 & 4-\lambda \end{bmatrix} \).

Step 2: Calculate the Determinant of A - λI

Find the determinant of the matrix \( A - \lambda I \):\[\text{det}(A - \lambda I) = \begin{vmatrix} 1-\lambda & -3 & 3 \ 3 & -5-\lambda & 3 \ 6 & -6 & 4 - \lambda \end{vmatrix} \].

Step 3: Expand the Determinant

Using the cofactor expansion on the first row, calculate the determinant:\[ \text{det}(A - \lambda I) = (1-\lambda)\begin{vmatrix} -5-\lambda & 3 \ -6 & 4-\lambda \end{vmatrix} + 3\begin{vmatrix} 3 & 3 \ 6 & 4-\lambda \end{vmatrix} - 3\begin{vmatrix} 3 & -5-\lambda \ 6 & -6 \end{vmatrix}\].

Step 4: Solve Individual 2x2 Determinants

Calculate each 2x2 sub-determinant:- \(\begin{vmatrix} -5-\lambda & 3 \ -6 & 4-\lambda \end{vmatrix} = (-5-\lambda)(4-\lambda) - (3)(-6) = \lambda^2 + \lambda - 30\)- \(\begin{vmatrix} 3 & 3 \ 6 & 4-\lambda \end{vmatrix} = (3)(4-\lambda) - (3)(6) = -3\lambda - 6\)- \(\begin{vmatrix} 3 & -5-\lambda \ 6 & -6 \end{vmatrix} = (3)(-6) - (-5-\lambda)(6) = 6\lambda + 12\)Substitute these in the expression from the previous step.

Step 5: Substitute and Simplify

Substitute the 2x2 determinants back into the cofactor expansion:\[\text{det}(A - \lambda I) = (1-\lambda)(\lambda^2 + \lambda - 30) + 3(-3\lambda - 6) - 3(6\lambda + 12)\],Expand and simplify the expression to find the characteristic polynomial:\[ = -\lambda^3 + \lambda^2 + 8\lambda - 48\].

Step 6: Find the Roots of the Polynomial

The roots of the polynomial \(-\lambda^3 + \lambda^2 + 8\lambda - 48 = 0\) are the eigenvalues of the matrix. Factor this polynomial or use a calculator to find that the roots are \( \lambda = 4, -2, -2 \).

Step 7: Confirm Eigenvalues Match Option

The calculated eigenvalues are \( 4, -2, -2 \), which corresponds to option (A). Therefore, the eigenvalues are \( 4, -2, -2 \).

Key Concepts

Characteristic Polynomial

The characteristic polynomial is a vital tool in linear algebra, especially when analyzing matrices. It provides a polynomial equation, derived from a matrix, which reveals important properties of the matrix. To find the characteristic polynomial of a matrix, you subtract a scalar \(\lambda\) from each of the elements on the principal diagonal of the matrix. The formula becomes \(|A - \lambda I|\), where \(A\) is the given matrix and \(I\) is the identity matrix of the same size as \(A\).

• This polynomial will be of degree \(n\), where \(n\) is the order of the square matrix.
• The coefficients of the polynomial give insight into the behavior and properties of the matrix.

Determining this polynomial is the first step towards finding the eigenvalues, which are crucial in solving systems of linear equations and understanding linear transformations.

Characteristic Equation

The characteristic equation arises from the characteristic polynomial. To find this equation, you set the characteristic polynomial equal to zero: \(|A - \lambda I| = 0\). Solving this equation provides the eigenvalues of the matrix, which are the roots of the equation.

• The characteristic equation aids in revealing the behavior of different matrix operations.
• It helps in applying diagonalization and other techniques for simplifying matrices.
• Research in this area can lead to understanding stability in systems influenced by the matrix.

Finding and solving the characteristic equation forms an essential aspect of studying matrix algebra and helps in gaining deeper insights into the structural properties of matrices.

Matrix Determinant

The determinant of a matrix is a scalar value that can be computed from its elements and reveals numerous properties about the matrix. When dealing with the matrix \(A - \lambda I\), determining its determinant leads to the characteristic polynomial.

• The determinant is crucial as it helps in resolving whether a matrix is invertible; if the determinant is non-zero, the matrix is invertible.
• In the context of eigenvalues, it plays a substantial role in the characteristic polynomial formation.
• The determinant also has geometric meanings, such as describing volume distortion during linear transformation.

Understanding the essence of the matrix determinant is fundamental to appreciating why certain vectors keep their direction through transformations and how matrices interact with space.

Spectrum of a Matrix

The spectrum of a matrix consists of all its eigenvalues, basically the set of all solutions to the characteristic equation. These eigenvalues provide valuable information about the matrix, applicable in numerous mathematical and engineering fields.

• The spectrum helps in predicting the behavior of dynamic systems.
• It reveals stability aspects of matrices when applied to various contexts.
• Normal and diagonal matrices, in particular, have spectra that reflect their simpler forms, making calculations more straightforward.

Thus, the spectrum provides a comprehensive picture of how certain actions and transformations are associated with the matrix and their subsequent implications in real-world scenarios.