Equivalent matrices are similar

Both are invertible, so each is P·(the other)·Q for suitable invertible P, Q — every invertible n×n matrix is equivalent to every other, because equivalence sees nothing finer than rank. But A has eigenvalue 1 and B has eigenvalue 2, and similarity preserves eigenvalues. Equivalence allows an independent change of basis in the domain and the codomain; similarity forces the same basis on both sides.