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Let $A \in \mathrm{GL}\,_n(\mathbb R)$ have integer entries. Let $b \in \mathbb R^n $ be a column vector also with integer entries. Then

  1. If $Ax = b$ , then entries of $x$ are also integers.

  2. if $Ax = b$ , then the entries of $x$ are rational.

  3. The matrix $A^{-1}$ has integer entries iff $\det(A) = \pm 1$.

For (1) is false by Cramer's Rule.

For (2) is true by Cramer Rule.

For (3), I think it is also true.

Thank you for sparing your valuable time in checking my solutions

André Nicolas
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user120386
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1 Answers1

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I think you are right. Since ${A^{ - 1}}$ has integer entries, $det({A^{-1}})=det(A)^{-1}$ is an integer. It's true only for neither $det(A)$ is 1 or -1. By the formula ${A^{ - 1}} = \frac{{{A^ * }}}{{\det \left( A \right)}}$, the other hand holds.

Junefi
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