Key to "Consider the system of equations described by {x1=2x1−3x2x2=4x1−5x2..."

High School
Algebra
Algebra I
Systems of equations

Jerold

Answered question

2021-09-09

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

Answer & Explanation

Velsenw

Skilled2021-09-10Added 91 answers

Consider the system of equations described by
${\begin{cases} x_{1} = 2 x_{1} - 3 x_{2} \\ x_{2} = 4 x_{1} - 5 x_{2} \end{cases}$
The system of equations in the matrix form
$(\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = [\begin{matrix} 2 & - 3 \\ 4 & - 5 \end{matrix}] (\begin{matrix} x_{1} \\ x_{2} \end{matrix})$
$\Rightarrow x^{'} = A x, w h e r e A = [\begin{matrix} 2 & - 3 \\ 4 & - 5 \end{matrix}], x = (x_{1}, x_{2})$
Now, $| A - λ I | = [\begin{matrix} 2 - λ & - 3 \\ 4 & - 5 - λ \end{matrix}] = 0$
$\Rightarrow λ^{2} + 3 λ + 2 = 0 \Rightarrow λ = - 2, - 1$
Hence, -1,-2 are the eigen values.
$I f, λ = - 1$
$T h e n [\begin{matrix} 3 & - 3 \\ 4 & - 4 \end{matrix}] (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix})$
$\Rightarrow 3 x_{1} - 3 x_{2} = 0 \Rightarrow x_{1} = x_{2} = t ϵ R$
Hence, $V_{1} = (\begin{matrix} 1 \\ 1 \end{matrix})$
$I f, λ = - 2$
$T h e n, [\begin{matrix} 4 & - 3 \\ 4 & - 3 \end{matrix}] (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix})$
$\Rightarrow 4 x_{1} - 3 x_{2} = 0$
$\Rightarrow x_{1} = \frac{3}{4} x_{2} = 0$
Hence, $V_{2} = (\begin{matrix} 1 \\ \frac{4}{3} \end{matrix})$
Hence, $V_{1} and V_{2}$ are the associated eigenvectors.

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