My only comment is why, unless this is a maths research thing :-)
Nobody would really invert a 2000x2000 matrix it would be faster to solve the equation and once solved about a million times faster to use that multiplying any input value by a 2000x2000 inverse matrix to get an answer.
So I am curious why anyone would do this.
Even on Matlab forums when people ask about large inverts they generally preface the answer with
You should ask yourself whether you really need to do the matrix inversion or not.