Solutions to systems of equations

Question

Suppose I have two systems of equations $\bf{Ax}$=$\bf{b}$ and $\bf{Cy}=\bf{d}$ each with unique solutions. What are the necessary and/or sufficient conditions for the solutions to be identical ?

Answer 1

We need that $A$ and $C$ have the same numbers of column vectors. If the solutions are unique the column vectors of the matrices are linearly independent, hence $A^tA$ and $C^tC$ are invertible. From here $$(A^tA)^{-1}A^tb=(C^tC)^{-1}C^td.$$