Check whether a system of linear equations has a a solution?

Questions
,
1

What is the fastest way to check whether a system of linear equations has a solution?

I know that LinearSolve can solve this problem, but if you don’t need to know the what the solution is, but only need to judge the existence, is there a more efficient method? The way to calculate rank seems to be slower.

Specifically, the equation can be given by the coefficient matrix A \in \mathbb{R}^{m\times n} and the column vector b, where m and n are large (at least thousands). They are both in the form of SparseArray.

2

A linear system .A.x - B == 0 is solvable if A and A extended by B to A B per row are equal

MatrixRank[A] ==     MatrixRank[ Transpose[ Append {Transpose[A],B ] ]

The background is that A , x = B is not solvable if the NullSpace[A] is not linear independenr on B.In this case impicitly one has one or more equations of the form 0 x = 1