##### 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 \(\lambda=-5\text{.}\)