Formula for $n$-dimensional parallelepiped

Question

What is the formula for the volume of a parallelepiped in $n$ dimensions with $v_1,v_2,\ldots,v_n$ as edges?

In an exercise it's given as $(\det[b_{i,j}])^{\frac12}$, where $b_{i,j}=v_i\cdot v_j$. But I also remember seeing somewhere that the volume is given by $|\det A|$, where $A$ is the matrix with columns $v_1,v_2,\ldots,v_n$.

Am I misunderstanding somewhere, or are these two values actually the same?

Answer 1

Suppose that $A = (v_1,\ldots,v_n)$, and you have the second formula for volume as $|\det A|$. Then the first is derived by having $\det(A^TA) = (\det A)^2$.