###### Example4.1.2

Consider the matrix $$A = \left[\begin{array}{rr} 7 \amp 6 \\ 6 \amp -2 \\ \end{array}\right]$$ and the vector $$\vvec=\twovec{2}{1}\text{.}$$ We find that

\begin{equation*} A\vvec = \left[\begin{array}{rr} 7 \amp 6 \\ 6 \amp -2 \\ \end{array}\right] \twovec{2}{1} = \twovec{20}{10} =10\twovec{2}{1} =10\vvec\text{.} \end{equation*}

In other words, $$A\vvec = 10\vvec\text{,}$$ which says that $$\vvec$$ is an eigenvector of the matrix $$A$$ with associated eigenvalue $$\lambda = 10\text{.}$$

Similarly, if $$\wvec = \twovec{-1}{2}\text{,}$$ we find that

\begin{equation*} A\wvec = \left[\begin{array}{rr} 7 \amp 6 \\ 6 \amp -2 \\ \end{array}\right] \twovec{-1}{2} = \twovec{5}{-10} =-5\twovec{-1}{2} =-5\wvec\text{.} \end{equation*}

Here again, we have $$A\wvec = -5\wvec$$ showing that $$\wvec$$ is an eigenvector of $$A$$ with associated eigenvalue $$\lambda=-5\text{.}$$

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