###### Example 4.1.2

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