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Avoiding redundancy in checking uniqueness of solution to linear equation

What I want is a function that given a matrix $A$ and vector $v$ will return a vector $x$ such that $Ax=v$ and will determine if this $x$ is unique or not, while doing a minimal amount of work. Currently I am using right nullity and the solve_right function separately, which I think results in some redundancy. Any advice on what I should do? There doesn't seem to be an obvious solution from the manual.