Linear isometry/Determinant is 1 or -1/Fact/Proof2

Proof

Due to fact, we have

${}{M^{\text{tr}}}M={\begin{pmatrix}1&0&\cdots &\cdots &0\\0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&1&0\\0&\cdots &\cdots &0&1\end{pmatrix}}\,.$

Therefore, the statement follows from the multiplication theorem for the determinant and from fact.

To fact