Find a value of k so that \displaystyle \left\lbrack \begin{array}{r} 1 \\ 1 \end{array} \right\rbrack e^{k t} is a solution to \displaystyle \boldsymbol{\vec{x}^{\,\prime}} = \left\lbrack \begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array} \right\rbrack \boldsymbol{\vec{x}}.