Chapter 3 - Determinants - 3.3 Properties of Determinants - 3.3 Exercises - Page 127: 77

The statement is correct.

Since A is an orthogonal matrix, then $A^{-1}=A^{T}$ and then $|A^{-1}|=|A^{T}|$ Since $A A^{-1}=I$, then $|A A^{-1}|=| A|| A^{-1} |=|I|=1$, therefore $| A^{-1} |=1/| A |$ We know that $| A^{T}|=| A |$. Thus $1/| A |=| A^{-1} |=|A^{T}|=| A |$ Therefore, we have $| A |^{2}=1$. This implies that either $| A |=1$ or $| A |=-1$