Exercise 3.3.7

Question

A real square matrix $Q$ is called orthogonal if $Q^\mathsf{T} Q = I$. Prove that if $Q$ is an orthogonal matrix then $\det Q = \pm 1$.

Jing Huang · Accepted Answer

\begin{proof}
$\det Q^\mathsf{T} Q = (\det Q^\mathsf{T}) (\det Q) = (\det Q)^2 = \det I = 1$. Thus $\det Q = \pm 1$.
\end{proof}

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