We call the vector x Rn integral if every component xi is an integer. Let A be a nonsingular n n matrix with integer entries. Prove that the system of equations Ax = b has an integral solution for every integral vector b Rn if and only if det A = 1. (Note that if A has integer entries, A maps integral vectors to integral vectors. When does A map the set of all integral vectors onto the set of all integral vectors?)

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