Each matrix over the commutative domain of main ideals may be reduced to diagonal form: \(D^A= U A V = {\text{diag}}(\mu_1^A,\dots,\mu_n^A)\), where \(\mu_i^A \) is the divisor of \(\mu_{i+1}^A.\) It is known that divisibility \(B | A\) leads to divisibility \(D^B | D^A\). The author investigates conditions under which the inverse implication holds true. Relations between divisibility and simultaneous reducibility of matrices \(A,B\) are ascertained.