Recursive 2x2 determinant
J, 21 25 bytes

0{0{(_2(_2-/ .*\|:)\])^:_


Usage:

   ]input=.(3,1,4,1),(5,9,2,6),(5,3,5,8),:(9,7,9,3)
3 1 4 1
5 9 2 6
5 3 5 8
9 7 9 3
   (0{0{(_2(_2-/ .*\|:)\])^:_) input
_1430


At every step we cut up the matrix to 2-by-2's and compute each ones determinant resulting in the next step's input matrix. We repeat this process until the result doesn't change (the final element is the determinant itself). We convert the final result to a scalar with 0{0{.

Try it online here.