# 2016-10 Factorization

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 $$\det B_1 = p_1^{e_1}, \det B_2 = p_2^{e_2}, \dots, \det B_k = p_k^{e_k}$$.

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