# What is the fastest way to check if matrix is invertible

I have a huge matrix that I suspect is not invertible.

What is the fastest function in mathematica to test it ?

I have remarked that MatrixRank is faster than NullSpace and Det, but is there an even faster method ?

Some precisions : it is a symbolic ~ 700*700 matrix full of number (not variables) that can be rationals.

• I think this may depend on the complexity of your rationals (e.g. size of the denominator). – mikado Jul 6 '17 at 19:13
• Why can't answers given in fast-method-to-check-if-a-matrix-is-singular or in what-is-the-most-efficient-way-to-determine-if-a-matrix-is-invertible be used? – Nasser Jul 6 '17 at 19:31
• @Nasser - these don't really deal with finding the rank of an exact matrix using Mathematica – mikado Jul 6 '17 at 21:17
• (The term usually used to describe such matrices is "exact numeric". (2) One can usually safely rule out rank deficiency with approximate methods, e.g. checking that the smallest singular value is not close to zero ("close" being relative to the precision used for the computation). (3) Using exact Det may well be generally the fastest direct method in Mathematica. – Daniel Lichtblau Jul 7 '17 at 0:31