Linear Algebra Calculator

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Example

Created on 2024-06-20Asked by Elizabeth Green (Solvelet student)

Find the eigenvalues of the matrix

A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}

Solution

To find the eigenvalues of the matrix

A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}

, we solve the characteristic equation

\det(A - \lambda I) = 0

. First, compute

A - \lambda I

A - \lambda I = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix}

Next, find the determinant:

\det(A - \lambda I) = \begin{vmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{vmatrix} = (4 - \lambda)(3 - \lambda) - (2 \cdot 1) = \lambda^2 - 7\lambda + 10

Solve the characteristic equation:

\lambda^2 - 7\lambda + 10 = 0

Factor the quadratic equation:

(\lambda - 2)(\lambda - 5) = 0

Thus, the eigenvalues are:

\lambda = 2 \quad \text{and} \quad \lambda = 5

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Avery Ramirez on Solvelet

1. Find the determinant of the matrix

A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}

.2. Determine whether the vectors

\mathbf{v} = [1, 2, 3]

and

\mathbf{w} = [2, -1, 1]

are linearly independent

DefinitionLinear Algebra by John D. Cook Linear algebra is the branch of mathematics dealing with linear equations, linear functions, and their representations through matrices and vector spaces. It consists of things like vector spaces, linear transformations, determinants, eigenvalues, and eigenvectors. Application : The one of the standard application which are generally solved is with the help of matrix 0peration is Solving System of Linear Equations, this is base use case of Linear Algebra.

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