A transformation with associated matrix \(\left( \begin{array} { r r r } m & 2 & 1 0 & 1 & - 2 2 & 0 & 3 \end{array} \right)\), where \(m\) is a constant, maps the vertices of a cube to points that all lie in a plane.
Find \(m\).
The transformations S and T of the plane have associated matrices \(\mathbf { M }\) and \(\mathbf { N }\) respectively, where \(\mathbf { M } = \left( \begin{array} { r r } k & 1 - 3 & 4 \end{array} \right)\) and the determinant of \(\mathbf { N }\) is \(3 k + 1\). The transformation \(U\) is equivalent to the combined transformation consisting of S followed by T .
Given that U preserves orientation and has an area scale factor 2, find the possible values of \(k\).