In the vector space of $2\times 2$ matrices, $M_{22}$, determine if the set $S$ below is linearly independent. \begin{equation*} S=\set{ \begin{bmatrix} 2&-1\\1&3 \end{bmatrix},\, \begin{bmatrix} 0&4\\-1&2 \end{bmatrix},\, \begin{bmatrix} 4&2\\1&3 \end{bmatrix} } \end{equation*}