Unique Solution of a Square System (Invertible Coefficient Matrix)

A square system $A\mathbf{x}=\mathbf{b}$ has exactly one solution for every right-hand side if and only if $A$ is invertible, i.e. $\det(A)\neq 0$ — so "for which parameter is the solution unique?" always reduces to solving $\det(A)\neq 0$.