matrices with determinant equals to one

Question

we already know what does it mean the determinant of a matrix is null, it's not invertible ! but what about matrices with determinant equals to $1$ ?! I know that the determinant of matrix is the volume of the polyhedron build from the rows of the matrix (or the columns) but what does it mean a unit volume, I mean is there any mathematical aspect represents this fact just like the invertibility ? Thank you for your time .

Answer 1

For a coordinate transformation $x_i \to x_i'$ the determinant of the matrix of the partial derivatives gives the correction factor for the volumes: $$ dV = dx_1 \cdots dx_n = \mbox{det } \left( \frac{\partial(x_1, \ldots x_n)}{\partial(x_1', \ldots, x_n')} \right) dx_1' \cdots dx_n' = \mbox{det }(J) \, dV' $$ where $J$ is called the Jacobian matrix, with $J_{ij} = \partial x_i/\partial x_j'$.

If $\mbox{det}(J) = 1$ then $dV = dV'$, so for these transformations the volumes are invariant.