# Consider the system of equations described byx_{1} = 2x_{1} - 3x_{2}x_{2} = 4x_{1} - 5x_{2}1. Write down the system of equations in matrix form.2. Find the eigenvalues of the system of equations.3. Find the associated eigenvectors.

Consider the system of equations described by
$$\begin{cases}x_1=2x_1-3x_2\\x_2=4x_1-5x_2\end{cases}$$
1. Write down the system of equations in matrix form.
2. Find the eigenvalues of the system of equations.
3. Find the associated eigenvectors.

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Velsenw

Consider the system of equations described by
$$\begin{cases}x_1=2x_1-3x_2\\x_2=4x_1-5x_2\end{cases}$$
The system of equations in the matrix form
$$\left(\begin{array}{c}x_{1}\\ x_{2}\end{array}\right)=\begin{bmatrix}2 & -3 \\4 & -5 \end{bmatrix} \left(\begin{array}{c}x_{1}\\ x_{2}\end{array}\right)$$
$$\Rightarrow x'=Ax,\ where\ A=\begin{bmatrix}2 & -3 \\4 & -5 \end{bmatrix}, x=(x_{1}, x_{2})$$
Now, $$|A-\lambda I|=\begin{bmatrix}2-\lambda & -3 \\4& -5-\lambda \end{bmatrix}=0$$
$$\displaystyle\Rightarrow\lambda^{{{2}}}+{3}\lambda+{2}={0}\Rightarrow\lambda=-{2},-{1}$$
Hence, -1,-2 are the eigen values.
$$\displaystyle{I}{f},\lambda=-{1}$$
$$Then \begin{bmatrix}3 & -3 \\4 & -4 \end{bmatrix} \left(\begin{array}{c}x_{1}\\ x_{2}\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$
$$\displaystyle\Rightarrow{3}{x}_{{{1}}}-{3}{x}_{{{2}}}={0}\Rightarrow{x}_{{{1}}}={x}_{{{2}}}={t}\epsilon\mathbb{R}$$
Hence, $$V_{1}=\left(\begin{array}{c}1\\ 1\end{array}\right)$$
$$\displaystyle{I}{f},\lambda=-{2}$$
$${T}{h}{e}{n},\begin{bmatrix}4 & -3 \\4 & -3 \end{bmatrix}\left(\begin{array}{c}x_{1}\\ x_{2}\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$
$$\displaystyle\Rightarrow{4}{x}_{{{1}}}-{3}{x}_{{{2}}}={0}$$
$$\displaystyle\Rightarrow{x}_{{{1}}}={\frac{{{3}}}{{{4}}}}{x}_{{{2}}}={0}$$
Hence, $$V_{2}=\left(\begin{array}{c}1\\ \frac{4}{3}\end{array}\right)$$
Hence,$$V_{1}\ \text{and}\ V_{2}$$ are the associated eigenvectors.