The Laplace Transform of a matrix of functions is simply the matrix of Laplace transforms of the individual elements. \[ \begin{align*} \mathscr{L}(B \textbf{x}+\textbf{g}) &= \mathscr{L}(B \textbf{x}...The Laplace Transform of a matrix of functions is simply the matrix of Laplace transforms of the individual elements. \[ \begin{align*} \mathscr{L}(B \textbf{x}+\textbf{g}) &= \mathscr{L}(B \textbf{x})+\mathscr{L}(\textbf{g}) \\[4pt] &= B \mathscr{L}(\textbf{x})+\mathscr{L}(\textbf{g}) \end{align*}\] We note that \((sI-B)^{-1}\) well defined except at the roots of the quadratic, \(s^{2}-4s+3\) determinant of \((sI-B)\) and is often referred to as the characteristic polynomial of \(B\).