TITLE: $|\text{det}(A)| = 1$ implies $A$ is orthogonal
QUESTION [6 upvotes]: I know that $A$ orthogonal $\Rightarrow$ |det($A$)| = 1. Now I need to prove or disprove the reversed statement:
$$
|\det(A)| = 1 \Rightarrow A \,\text{ is orthogonal}
$$
This is what I'm currently trying:
$$
|\det(A)| = 1 \Rightarrow \det(A)^2 = 1 \Rightarrow \det(AA^t) = 1
$$
But I'm unsure whether this implies, that $AA^t = E_n$. Any help is welcome at this point. Maybe the statement isn't even true.

REPLY [5 votes]: No. Take any matrix $A$ with determinant $d\not=0$ and divide the elements of the first row by $d$. Then the new matrix has determinant $1$. Now it should be easy to find a counterexample to your implication.