### Problem 10: Eigenvalues and Eigenvectors**Objective:** Determine the eigenvalues and eigenvectors of the given matrix.**Matrix:**\[\begin{pmatrix}2 & 4 \\3 & -2\end{pmatrix}\]### Explanation:To find the eigenvalues of a matrix, we solve the characteristic equation:\[\text{det}(A - \lambda I) = 0\]where $ A $ is the given matrix and $ I $ is the identity matrix of the same dimension. In this case, we have:\[A - \lambda I = \begin{pmatrix}2-\lambda & 4 \\3 & -2-\lambda\end{pmatrix}\]Finding the determinant:\[\text{det}(A - \lambda I) = (2-\lambda)(-2-\lambda) - (4 \times 3)\]Simplifying this further gives the characteristic polynomial which can be solved for $ \lambda $, the eigenvalues.Once the eigenvalues are found, substitute each eigenvalue back into:\[(A - \lambda I)x = 0\]to find the corresponding eigenvectors $ x $.

Consider the matrix A=243−2 First we will find eigen value of the matrix A The eigenvalues of A are…

Answered: Determine the eigen values/vectors of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### Problem 10: Eigenvalues and Eigenvectors

**Objective:** Determine the eigenvalues and eigenvectors of the given matrix.

**Matrix:**
\[
\begin{pmatrix}
2 & 4 \\
3 & -2
\end{pmatrix}
\]

### Explanation:

To find the eigenvalues of a matrix, we solve the characteristic equation:

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

where $ A $ is the given matrix and $ I $ is the identity matrix of the same dimension. In this case, we have:

\[
A - \lambda I =
\begin{pmatrix}
2-\lambda & 4 \\
3 & -2-\lambda
\end{pmatrix}
\]

Finding the determinant:

\[
\text{det}(A - \lambda I) = (2-\lambda)(-2-\lambda) - (4 \times 3)
\]

Simplifying this further gives the characteristic polynomial which can be solved for $ \lambda $, the eigenvalues.

Once the eigenvalues are found, substitute each eigenvalue back into:

\[
(A - \lambda I)x = 0
\]

to find the corresponding eigenvectors $ x $.

Expert Solution

Step 1

Consider the matrix $A = (\begin{matrix} 2 & 4 \\ 3 & - 2 \end{matrix})$

First we will find eigen value of the matrix $A$

The eigenvalues of $A$ are the roots of the characteristic equation $d e t (A - λ I) = 0$

Hence,

$\begin{matrix} \det (A - λ I) = 0 \\ \det ((\begin{matrix} 2 & 4 \\ 3 & - 2 \end{matrix}) - λ (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})) = 0 \\ \det ((\begin{matrix} 2 & 4 \\ 3 & - 2 \end{matrix}) - (\begin{matrix} λ & 0 \\ 0 & λ \end{matrix})) = 0 \\ \det ((\begin{matrix} 2 - λ & 4 \\ 3 & - 2 - λ \end{matrix})) = 0 \end{matrix}$

Now,

$\begin{matrix} (2 - λ) (- 2 - λ) - 12 = 0 \\ - 4 - 2 λ + 2 λ + λ^{2} = 0 \\ λ^{2} - 4 = 0 \\ λ = 4 and - 4 \end{matrix}$

Therefore, eigenvalues of the matrix $A = (\begin{matrix} 2 & 4 \\ 3 & - 2 \end{matrix})$ is $λ_{1} = 4 and λ_{2} = - 4$ .

Step 2

Now, we will find the eigenvector corresponding to eigenvalues $λ_{1} = 4 and λ_{2} = - 4$ of the given matrix $A$

We know that the eigenvector corresponding to every eigenvalue $λ$ is given by $(A - λ I) x = 0$

Hence, the eigenvector corresponding to every eigenvalue $λ_{1} = 4$ is given by $(A - 4 I) x = 0$

Now, solve $(A - 4 I) x = 0$

$\begin{matrix} (A - 4 I) x = 0 \\ ((\begin{matrix} 2 & 4 \\ 3 & - 2 \end{matrix}) - 4 (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})) (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) \\ ((\begin{matrix} 2 & 4 \\ 3 & - 2 \end{matrix}) - (\begin{matrix} 4 & 0 \\ 0 & 4 \end{matrix})) (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) \\ (\begin{matrix} - 2 & 4 \\ 3 & - 6 \end{matrix}) (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) \\ (\begin{matrix} - 2 x_{1} + 4 x_{2} \\ 3 x_{1} - 6 x_{2} \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) \end{matrix}$

Hence, $\{\begin{array}{l} - 2 x_{1} + 4 x_{2} = 0 \\ 3 x_{1} - 6 x_{2} = 0 \end{array}\} ..... (1)$

Now, solve the system (1), we get

$x_{1} = 2 and x_{2} = 1$

Hence, the eigenvector corresponding to every eigenvalue $λ_{1} = 4$ is $x = (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 2 \\ 1 \end{matrix})$

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