Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?

From Santha and Tan, "Verifying the determinant in parallel":
Other notes: Matrix multiplication, inversion, and determinantfinding are all equally complex. I will quote from Kaltofen and Villard, "Computing the sign or the value of the determinant of an integer matrix, a complexity survey":
Since unimodularity amounts to invertibility, your question is equivalent to "is determining invertibility easier than actually computing the inverse (if it exists)?" 

