# Orthogonal matrix with determinant $-1$

I must provide a counterexample for the statement

If $A$ is an orthogonal matrix, then $\det(A)=1$.

I know that an orthogonal matrix may have a determinant of $-1$, but how can I find such a matrix?

• Hint: Reflection – Hayden May 2 '15 at 16:35
• How abou tthe $1\times 1$ matrix $(-1)$? – Hagen von Eitzen May 2 '15 at 16:36
• @HagenvonEitzen probably the best counter example :) – Sebastian Bechtel May 2 '15 at 16:42

$$\begin{pmatrix}1&0\\0&-1 \end{pmatrix}$$ in an orthonormal basis.
Consider the space $\Bbb R$, and the application $$f(x) = -x.$$
Take the identity matrix and interchange two columns. The resulting matrix will have determinant $-1$ and will be orthogonal.