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I have a doubt: I know that if a matrix is orthogonal, then it's determinant is either 1 or -1. What I'd like to know is if having a determinant equal to either 1 or -1 necessarily implies that a matrix is orthogonal.

Thank you!

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    $\begingroup$ No. ${}{}{}{}{}$ $\endgroup$ – user296602 Mar 24 '16 at 15:52
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Just consider $\begin{pmatrix} 1&2\\0&1 \end{pmatrix}$.

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Most certainly not. Take any invertible matrix $A\in\mathbb R^{n\times n}$ where $n\geq 2$. Now, take the matrix

$$B=\frac{1}{\sqrt[n]{|\det(A)|}}\cdot A$$

Then the determinant of $B$ is $1$ or $-1$, but $B$ doesn't need to be an orthogonal matrix, since if $BB^T=B^TB=I$, that would mean that $A^TA$ is also a scalar multiple of $I$, and this is surely not the case for all matrices!

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