Answer to Question #235230 in Algebra for Ash

Question

Find the eigenvalues and corresponding eigenvectors of matrix G given below.

G= [2 -3]

[4 -5]

Answer 1

$G= \begin{bmatrix} 2 & -3 \\ 4 & -5 \end{bmatrix}$

Finding the eigenvalues:

$\det \left(\begin{pmatrix}2&-3\\ 4&-5\end{pmatrix}-λ\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\right)$

$=\begin{pmatrix}2-λ&\left(-3\right)-0\\ 4-0&\left(-5\right)-λ\end{pmatrix}$

$\left(2-λ\right)\left(-5-λ\right)-\left(-3\right)\cdot \:4$

$λ^2+3λ+2=0$

Solving this $λ^2+3λ+2=0$

$λ=-1,\:λ=-2$

Therefore eigenvalues are: -1 and -2

Calculating Eigenvectors for $: λ=-1$

$= \begin{pmatrix}2&-3\\ 4&-5\end{pmatrix}-\left(-1\right)\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$

$=\begin{pmatrix}2&-3\\ 4&-5\end{pmatrix}-\begin{pmatrix}-1&0\\ 0&-1\end{pmatrix}$

$=\begin{pmatrix}3&-3\\ 4&-4\end{pmatrix}$

To solve $\begin{pmatrix}3&-3\\ 4&-4\end{pmatrix} \begin{pmatrix} x\\y\end{pmatrix} = \begin{pmatrix} 0\\0\end{pmatrix},$ reduce the matrix

$= \begin{pmatrix} 1\\1 \end{pmatrix}$ as the eigenvectors for $λ=-1$

Calculating Eigenvectors for $: λ=-2$

$= \begin{pmatrix}2&-3\\ 4&-5\end{pmatrix}-\left(-2\right)\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$

$=\begin{pmatrix}4&-3\\ 4&-3\end{pmatrix}$

To solve $\begin{pmatrix}4&-3\\ 4&-3\end{pmatrix} \begin{pmatrix} x\\y\end{pmatrix} = \begin{pmatrix} 0\\0\end{pmatrix},$ reduce the matrix

$= \begin{pmatrix} 3\\4 \end{pmatrix}$ as the eigenvectors for $λ=-2$

The Eigenvectors for $\begin{pmatrix}2&-3 \\ 4 & -5\end{pmatrix}$ becomes $=\begin{pmatrix}1\\ 1\end{pmatrix},\:\begin{pmatrix}3\\ 4\end{pmatrix}$