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

Step-by-Step Solution

Answer

The eigenvalues are \(4, -2, -2\), which matches option (A).

Step 1: Define the Characteristic Polynomial

The characteristic polynomial of a matrix \( A \) is given by \( |A - \lambda I| \), where \( \lambda \) is a scalar and \( I \) is the identity matrix. For a 3x3 matrix \( A \), \( |A - \lambda I| \) will be a cubic polynomial in \( \lambda \).

Step 2: Set Up the Characteristic Matrix

For the matrix \( A = \begin{bmatrix} 1 & -3 & 3 \ 3 & -5 & 3 \ 6 & -6 & 4 \end{bmatrix} \), the characteristic matrix \( A - \lambda I \) is \( \begin{bmatrix} 1-\lambda & -3 & 3 \ 3 & -5-\lambda & 3 \ 6 & -6 & 4-\lambda \end{bmatrix} \).

Step 3: Compute the Determinant

Calculate the determinant of the matrix \( A - \lambda I \):\[|A - \lambda I| = \begin{vmatrix} 1-\lambda & -3 & 3 \ 3 & -5-\lambda & 3 \ 6 & -6 & 4-\lambda \end{vmatrix}\]Using the cofactor expansion along the first row, compute the determinant by using the formula for a 3x3 matrix.

Step 4: Solve the Determinant Step-by-Step

Expand \(|A - \lambda I|\) using cofactor expansion:\[|A - \lambda I| = (1-\lambda) \cdot \begin{vmatrix} -5-\lambda & 3 \ -6 & 4-\lambda \end{vmatrix} + 3 \cdot \begin{vmatrix} 3 & 3 \ 6 & 4-\lambda \end{vmatrix} + 3 \cdot \begin{vmatrix} 3 & -5-\lambda \ 6 & -6 \end{vmatrix}\]Calculate each 2x2 determinant and sum them up.

Step 5: Expand Each Minor and Simplify

1. Calculate \( \begin{vmatrix} -5-\lambda & 3 \ -6 & 4-\lambda \end{vmatrix} = (-5-\lambda)(4-\lambda) - (-18) \)2. Calculate \( \begin{vmatrix} 3 & 3 \ 6 & 4-\lambda \end{vmatrix} = 3(4-\lambda) - 18 \)3. Calculate \( \begin{vmatrix} 3 & -5-\lambda \ 6 & -6 \end{vmatrix} = -18 + 6(-5-\lambda) \)Next, insert these results back into the determinant and simplify to find the polynomial.

Step 6: Formulate the Characteristic Polynomial

Combine all minors calculated in the previous step:\[|A - \lambda I| = (1-\lambda)[(-5-\lambda)(4-\lambda) + 18] + 3[12 - 3\lambda - 18] + 3[-18 - 6(-5-\lambda)]\]Simplify to get a polynomial in terms of \( \lambda \).

Step 7: Solve the Characteristic Equation

The characteristic equation is obtained by setting the characteristic polynomial equal to zero: \( |A - \lambda I| = 0 \).Solve this equation to find the values of \( \lambda \), which are the eigenvalues of matrix \( A \).

Step 8: Compare Eigenvalues with Given Options

After solving the characteristic equation, compare the obtained eigenvalue set with the given options (A) through (D) to identify the correct choice.

Key Concepts

Eigenvalues

Eigenvalues are among the core concepts in matrix algebra. They are special numbers associated with a matrix that give insight into its properties. Specifically, for a square matrix \( A \), an eigenvalue \( \lambda \) is a scalar such that there exists a non-zero vector \( \mathbf{v} \) satisfying the equation:

• \( A\mathbf{v} = \lambda \mathbf{v} \)

This equation implies that applying the matrix \( A \) to the vector \( \mathbf{v} \) is the same as scaling \( \mathbf{v} \) by \( \lambda \). Understanding eigenvalues helps us in many ways, such as determining the stability of systems in engineering or the long-term behavior of a transition matrix in stochastic processes.

• Important note: the collection of all eigenvalues of a matrix is known as the spectrum of that matrix.

Determinant

The determinant is a numerical value that can be calculated from a square matrix, and it provides key insights into the matrix’s properties. A determinant, denoted as \( |A| \) for a matrix \( A \), assists in determining several vital matrix attributes. For instance:

• If \( |A| = 0 \), the matrix is singular, implying it doesn’t have an inverse.
• The determinant's sign tells whether the matrix operation preserves orientation (positive determinant) or reverses it (negative determinant).
• For a 3x3 matrix, the determinant calculation involves a specific arithmetic operation on its elements, using a method called the cofactor expansion.

In the context of eigenvalues, a fascinating relationship exists: the product of the eigenvalues of a matrix equals the determinant of the matrix. This relationship highlights how eigenvalues encapsulate pivotal information about the matrix.

Characteristic Equation

The characteristic equation is a fundamental equation in matrix algebra that emerges from the characteristic polynomial. It is used to find the eigenvalues of a matrix. For a given \( n\times n \) matrix \( A \), the characteristic equation is derived by setting the determinant of \( A - \lambda I \) to zero:

• \(|A - \lambda I| = 0\)

Here, \( \lambda \) represents a scalar (potential eigenvalues), while \( I \) is the identity matrix of similar dimensions as \( A \). Solving this equation, typically a polynomial of degree \( n \), yields the eigenvalues of the matrix. Each solution \( \lambda \) corresponds to an eigenvalue of the matrix \( A \).

• The characteristic equation is integral to understanding the fundamental nature and dynamics of linear transformations represented by matrices.

Matrix Algebra

Matrix algebra is the branch of mathematics focusing on the study of matrices and their operations. It provides the fundamental tools for expressing and solving systems of linear equations, transformations, and various other complex operations in mathematics and applied sciences.

• Basic operations include addition, subtraction, and multiplication of matrices, alongside finding determinants and inverses.
• Matrix multiplication is particularly crucial as it corresponds to composite transformations in geometric contexts.
• In the realm of eigenvalues and eigenvectors, matrix algebra serves as the primary framework to analyze how matrix transformations affect vector spaces.

Through matrix algebra, we can interpret complex concepts in a structured and methodical manner, making it invaluable not only in pure mathematics but also in fields such as physics, computer graphics, and engineering.