Inverse and determinant of square matrix

The inverse (AI) and determinant (det) of a given square matrix (AO) may be directly found by
[AI,det] = inv1(AO)
It uses automatic pivoting scheme. All computations involves only simple matrix multiplication.

The direct result without pivoting may also be found by
AI = inv0(AO)
The sourse code is only 4 statement lines. Yet it works for AO = randn(n), even n = 1000.
However, it fails for some simple peculiar matrix, such as AO = [0 1; 1 0].