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

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.