Chapter 3.3, Problem 65E

Guided Proof Prove that the determinant of an invertible matrix $A$ is equal to $\pm 1$ when all of the entries of $A$ and $A^{-1}$ is integers.

Getting Started: Denote $\det \left(A\right)$ as $x$ and $\det \left(A^{-1}\right)$ as $y$. Note that $x$ and $y$ are real numbers. To prove that $\det \left(A\right)$ is equal to $\pm 1$, you must show that both $x$ and $y$ are integers such that their product $xy$ is equal to 1.

$\left(\text{i}\right)$ Use the property for the determinant of a matrix product to show that $xy=1$.

$\left(\text{ii}\right)$ Use the definition of a determinant and the fact that the entries of $A$ and $A^{-1}$ are integers to show that both $x=\det \left(A\right)$ and $y=\det \left(A^{-1}\right)$ are integers.

$\left(\text{iii}\right)$ Conclude that $x=\det \left(A\right)$ must be either $1$ or $-1$ because these are the only integer solutions to the equation $xy=1$