Prove that every orthogonal matrix $(Q^T Q = I)$ has determinant $1$ or $-1$.

Question

A matrix $Q$ is orthogonal if $Q^TQ=I$. I want to prove that if $Q$ is an orthogonal matrix, then $|Q|=\pm 1$. My try is : $$ |Q||Q| = |I|$$ $$|Q||Q| = 1$$

Answer 1

The proof is indeed correct, with the correct identification that $\mathrm{det}(Q^T)=\mathrm{det}(Q)$. As requested in chat, a fleshed out proof of the same

\begin{align*} Q^TQ=I&\Rightarrow \mathrm {det}(Q^TQ)=\mathrm {det}(I)\\ &\Rightarrow \mathrm {det}(Q^T)\mathrm {det}(Q)=1\\&\Rightarrow \mathrm {det}(Q)\mathrm {det}(Q)=1\\&\Rightarrow (\mathrm {det}(Q))^2=1\\&\Rightarrow \mathrm {det}(Q)=\pm 1 \end{align*}