Yes. Because C=E*A for some matrix E.
actually this is the fundamental reason row equivalence is a useful notion in solving linear systems.
i.e. this says exactly that any solutions of the equation AX=0, also solve CX=0.
recall the use of row reduction in solving systems: you reduce A to an equivalent matrix such that CX=0 is easy to solve. then you announce that the solutions are also solutions of AX=0.