Exercise 20

The algebraic property of determinants that factoring a scalar out of a single row will multiply the determinant by that scalar shows that where $H$ is $3\!\times\!3$, the determinant of $cH$ is $c^{3}$ times the determinant of $H$. Explain this geometrically, that is, using Theorem 1.5. (The observation that increasing the linear size of a three-dimensional object by a factor of $c$ will increase its volume by a factor of $c^{3}$ while only increasing its surface area by an amount proportional to a factor of $c^{2}$ is the Square-cube law .)