Recovering an eigenvector from its eigenvalue

MediumLinear Algebra~4m

The matrix $A=\begin{pmatrix}4&3\\2&3\end{pmatrix}$ has an eigenvalue $\lambda=6$ . The corresponding eigenvector can be written in the form $(x,\,2)$ . What is $x$ ?

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Solve $(A-6I)v=0$ ; the two rows give the same ratio between $v_1$ and $v_2$ .
The direction is $(3,2)$ ; rescale so the second component is $2$ .

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Recovering an eigenvector from its eigenvalue

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