2x2 Matrix Eigenvector Tool

A =

Check with WolframAlpha. Link is updated for

A

Based on the post

2 \times 2

Eigenvalues

Characteristic polynomial

p (λ) = λ^{2} - 7 λ + 6

Eigenvalues

\begin{aligned} λ_{1} & = 6 \\ λ_{2} & = 1 \end{aligned}

\begin{aligned} x_{1} & = (\begin{matrix} 2 \\ 1 \end{matrix}) \\ x_{2} & = (\begin{matrix} - 3 \\ 1 \end{matrix}) \end{aligned}

Work

\begin{aligned} x_{1} & = (\begin{matrix} a - d + \sqrt{(a - d)^{2} + 4 b c} \\ 2 c \end{matrix}) \\ = (\begin{matrix} - 1 + \sqrt{(- 1)^{2} + 4 (1) (6)} \\ 2 (1) \end{matrix}) = (\begin{matrix} 4 \\ 2 \end{matrix}) \\ x_{1} & = \frac{1}{2} \cdot (\begin{matrix} 4 \\ 2 \end{matrix}) = (\begin{matrix} 2 \\ 1 \end{matrix}) \end{aligned}

\begin{aligned} x_{2} & = (\begin{matrix} a - d - \sqrt{(a - d)^{2} + 4 b c} \\ 2 c \end{matrix}) \\ = (\begin{matrix} - 1 - \sqrt{(- 1)^{2} + 4 (1) (6)} \\ 2 (1) \end{matrix}) = (\begin{matrix} - 6 \\ 2 \end{matrix}) \\ x_{2} & = \frac{1}{2} \cdot (\begin{matrix} - 6 \\ 2 \end{matrix}) = (\begin{matrix} - 3 \\ 1 \end{matrix}) \end{aligned}