Existence and Uniqueness of Solutions for Ax

Assume that A is an n x n square matrix. Then the following statements can be made  

Admits the unique trivial solution x = 0 if and only if rank(A) = n 

Admits an (n - r)-parameter family of nontrivial solutions, in addition to the trivial solution, if and only if rank(A) = r n 

Stated more succinctly, if A is of full rank and b is independent of any columns of A, then a unique solution exists otherwise, either no solution exists or an infinite number of solutions exist. 

MMULT is the only function that operates on both matrices and vectors. This function requires the two arrays to be conformable. 

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