Suppose that \( A \) is an \( n \times n \) matrix with integer entries and \( \det A = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k} \) for primes \( p_1, p_2, \dots, p_k \) and positive integers \( e_1, e_2, \dots, e_k \). Prove that there exist \( n \times n \) matrices \( B_1, B_2, \dots, B_k \) with integer entries such that \( A = B_1 B_2 \dots B_k \) and \( \det B_1 = p_1^{e_1}, \det B_2 = p_2^{e_2}, \dots, \det B_k = p_k^{e_k} \).