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.

Jerold 2021-09-09 Answered

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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Expert Answer

Velsenw
Answered 2021-09-10 Author has 13531 answers

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.

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