Consider a matrix A, and let B = rref(A). Is ker(A) necesarily equal to ker(B), and is im(A) necessarily equal to im(B)? I want to say that the answer to both questions are yes because A and B are the same matrix, i.e. there are a finite number of elementary operations that can change A to B, and vice versa. Therefore, if they are the same matrices, then they necessarily will have the same kernel and image as each other. Is my reasoning correct?