system of equations when the matrix corresponding $\det(A)=\pm1$ has integers solution

Question

I am reading a book about continued fractions and one of the theorem's proof constructs a system of linear equations and states that the matrix corresponding with the system of equations satisfies $\det(A)=\pm1$ and hence has integers solution why is that so?

Answer 1

I think it follows from Cramer's rule. If you have a system of linear equations in the form $Ax=b$ such that $A$ is nonsingular (or equivalently $\det(A)\neq 0$), then we can solve it by Cramer's rule. In particular, we have $x_i=\frac{\det B_i}{\det A}$ for each $i$. Therefore, if $b$ has integer entries and $\det A=\pm 1$, then $\det B_i$ is integer and as a result $x_i$ is integer.

Answer 2

Note that $\mathrm{adj}(A) \in M_{n\times n}(\Bbb Z)$ if $A \in M_{n\times n}(\Bbb Z)$, and $\mathrm{adj}(A) A = \det(A) I_n$, so $A^{-1} = (\det(A))^{-1} \mathrm{adj}(A) \in M_{n\times n}(\Bbb Z)$.

$$Ax=b$$ $$x=A^{-1}b \in M_{n\times 1}(\Bbb Z)$$