Another application of determinants:
- Compute a vector perpendicular to other two vectors.
- Compute the rotational of a vector field
- Determine if three vector are linearly dependent
- If you consider a complex number z = a + bi, then there is a linear transformation from C to C given by x -> xz, than you can see as a linear transformation from R^2 to R^2, the determinant of that matrix is the amplifying factor for the area of figures.