Answer
$-28$
Work Step by Step
We find the determinant by performing a series of elementary row exchanges, leading to an echelon matrix whose determinant equals the determinant of the original matrix.
$\begin{vmatrix}1&-1&-3&0\\0&1&5&4\\-1&0&5&3\\3&-3&-2&3\end{vmatrix}=\begin{vmatrix}1&-1&-3&0\\0&1&5&4\\0&-1&2&3\\0&0&7&3\end{vmatrix}=\begin{vmatrix}1&-1&-3&0\\0&1&5&4\\0&0&7&7\\0&0&7&3\end{vmatrix}=\begin{vmatrix}1&-1&-3&0\\0&1&5&4\\0&0&7&7\\0&0&0&-4\end{vmatrix}=1\cdot 1\cdot 7\cdot (-4)=-28$
