Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 4

The parallelepiped volume is the absolute value of the scalar triple product of its edge vectors, equivalently the absolute determinant of the matrix with those columns. Expansion along the first row givesTherefore

Multiplying a three-dimensional matrix by a real $\lambda$ multiplies its determinant by ${\lambda }^3$, whereas multiplicativity of the determinant gives $\det (A^5)=(\det A{)}^5$. Therefore equality of the two parallelepiped volumes meansBoth real signs are allowed:A negative value reverses the oriented volume while preserving the ordinary volume. If a positive scale factor is intended, select the positive sign.