In the space 퐿₂ ( ℝ d ; ℂ n ), a second order elliptic matrix differential selfajoint operator ℬ , 0<휀 ≤ 1, is considered. The senior part of the operator is presented in a factored form, the operator involves also terms of fiest and zero orders. The operator ℬ is positive definite, its coefficients are periodic and depend on 혅/휀. The limit behavior (as the periodic tends to zero) of the operator exponential 푒 -ℬ , ≥ 0 is studied. An approximation for this exponential in the operator (퐿 2 → 퐿 2 )-norm is found with an error estimate of order 푂(휀 2 ). The corrector is taken into account. The results are applied to the homogenizations of the Cauchy problem for parabolic systems.