Does Numpy/Python automatically detect the presence of GPU and utilize
it to speed up matrix computation (e.g. numpy.multiply,
numpy.linalg.inv, … etc)?
No.
Or do I have code in a specific way to exploit the GPU for fast
computation?
Yes. Search for Numba, CuPy, Theano, PyTorch or PyCUDA for different paradigms for accelerating Python with GPUs.