Answer
$0$
Work Step by Step
We reduce the matrix to an echelon form by a series of row replacement operations.
$\begin{vmatrix}1&3&0&2\\-2&-5&7&4\\3&5&2&1\\1&-1&2&-3\end{vmatrix}=\begin{vmatrix}1&3&0&2\\0&1&7&8\\0&-4&2&-5\\0&0&0&0\end{vmatrix}=\begin{vmatrix}1&3&0&2\\0&1&7&8\\0&0&30&27\\0&0&0&0\end{vmatrix}=1\cdot 1\cdot 30\cdot 0=0$
Note that the answer was clear as soon as elementary row operations produced a row of all zeros.
