Proof that the absolute determinant equals the product of norms of orthogonal vectors

Question

$A$ is a $n\times n$ matrix of $n$ orthogonal vectors $x_1, \dots, x_n$ in $\mathbb{R}^n$ that are not necessarily normalized. Prove that the absolute determinant equals the product of vector norms

$$|\det(A)| = \prod_{k=1}^n |x_k|.$$

Answer 1

The matrix $$\begin{bmatrix} \frac{x_1}{\|x_1\|} & \frac{x_2}{\|x_2\|} & \cdots &\frac{x_n}{\|x_n\|}\end{bmatrix}$$ has orthonormal columns so it is orthogonal. Therefore its determinant is $\pm 1$ so $$\pm 1 = \det \begin{bmatrix} \frac{x_1}{\|x_1\|}& \cdots& \frac{x_n}{\|x_n\|}\end{bmatrix} = \frac1{\|x_1\|\cdots \|x_n\|} \det \begin{bmatrix} x_1 & \cdots & x_n\end{bmatrix} = \frac{\det A}{\|x_1\|\cdots \|x_n\|}.$$

We conclude $$\left|\det A\right| = \|x_1\|\cdots \|x_n\|.$$