Differential Equations

if i have the general solution of a Differential Equation as:
\begin{equation*}
\bold{x}(t) = \left[ \begin {array}{c} u\\\noalign{\medskip}v\\\noalign{\medskip}w
\end {array} \right] = c_1\left[ \begin {array}{c} 1\\\noalign{\medskip}2\\\noalign{\medskip}1
\end {array} \right] e^{t} + (c_2 + c_3t)\left[ \begin {array}{c} -1\\\noalign{\medskip}0\\\noalign{\medskip}1
\end {array} \right] e^{-t} + c_3\left[ \begin {array}{c} -1\\\noalign{\medskip}1\\\noalign{\medskip}0
\end {array} \right] e^{-t}
\end{equation*}

.. if u have latex plug it in there ... frigg i cant get the code to work
How can i prove this formula includes all possible solutions.. Im pretty sure its using the Wronskian test but im not sure how to do for this! Any help?!

Edit it and use [ math ] and [ /math ] not "begin{equation*}" and "end{equation*}".

I found a site that explains the wronskian determinant for vectors, thanks i think i figured out the matrix thing.