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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?

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  • $\begingroup$ Hint: Reflection $\endgroup$ – Hayden May 2 '15 at 16:35
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    $\begingroup$ How abou tthe $1\times 1$ matrix $(-1)$? $\endgroup$ – Hagen von Eitzen May 2 '15 at 16:36
  • $\begingroup$ @HagenvonEitzen probably the best counter example :) $\endgroup$ – Sebastian Bechtel May 2 '15 at 16:42
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$$\begin{pmatrix}1&0\\0&-1 \end{pmatrix}$$ in an orthonormal basis.

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Consider the space $\Bbb R$, and the application $$ f(x) = -x. $$

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Take the identity matrix and interchange two columns. The resulting matrix will have determinant $-1$ and will be orthogonal.

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