Eigenvalues Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Harper Martinez (Solvelet student)

Find the eigenvalues of the matrix

A = \begin{bmatrix} 3 & -1 \\ -1 & 3 \end{bmatrix}

Solution

To find the eigenvalues of the matrix

A = \begin{bmatrix} 3 & -1 \ -1 & 3 \end{bmatrix}

:Step 1: Write down the characteristic equation:

\text{det}(A - \lambda I) = 0,

where

I

is the identity matrix. Step 2: Substitute the values of

A

into the characteristic equation:

\text{det}\left(\begin{bmatrix} 3-\lambda & -1 \ -1 & 3-\lambda \end{bmatrix}\right) = (3-\lambda)^2 - (-1)(-1) = \lambda^2 - 6\lambda + 8 = 0.

Step 3: Solve the characteristic equation to find the eigenvalues:

(\lambda - 4)(\lambda - 2) = 0 \implies \lambda_1 = 4, \quad \lambda_2 = 2.

Step 4: Conclusion. The eigenvalues of

A

are

\lambda_1 = 4

and

\lambda_2 = 2

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by William Rodriguez on Solvelet

1. Find the eigenvalues of the matrix

A = \begin{bmatrix} 2 & 1 \ 1 & 2 \end{bmatrix}

.2. Determine whether the matrix

B = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}

has distinct eigenvalues.

DefinitionAn Eigenvalue is a scalar value that represents a scaling factor in the linear equations represented by matrices. Given a matrix A, an eigenvalue λ and its corresponding eigenvector v are defined by Av = λv. Eigenvalues describe how a matrix stretches, compresses or rotates vectors in the direction of its eigenvectors. Eigenvalues determine the stability, or otherwise, of systems in linear algebra, differential equations and physics - they provide information concerning the dynamics of a system and its equilibrium points. For Example EIGENVALUES OF A 2x2 MATRIX. the eigenvalues of a 2x2 matrix [acbd], are the solutions to the characteristic equation λ2−(a+d)λ+(ad−bc)=0.

Related Calculators

Vector Operations Calculator Method of Lines Calculator Arc Length Calculator Multiplication of decimals Calculator

Need topic explanation ? Get video explanation