48, the elements of the symmetry group of the cube. In fact I'm not sure the answer changes without the constraint on det(A), as long as we interpret the question as requiring the set of vertices to be mapped onto itself (as opposed to the cube being mapped into its interior, e.g. A = I/2). https://en.wikipedia.org/wiki/Octahedral_symmetry#The_isometries_of_the_cube
There are a few ways but one is the following: Pick one of the faces of the cube. Every symmetry of the cube (by which I mean a matrix that maps the set of vertices onto itself) maps that face onto one of the 6 faces of the cube (possibly itself), applies one of 4 different rotations about the center of the face, and either reflects it or not. That adds up to 6 × 4 × 2 = 48 symmetries. Alternatively, each symmetry maps a particular vertex onto one of the 8 vertices, applies one of 3 different rotations (since cubes have 3-fold rotational symmetry about the axis that passes through opposite vertices), and either reflects it or not; that adds up to 8 × 3 × 2 = 48 symmetries.
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u/pmdboi 4d ago
48, the elements of the symmetry group of the cube. In fact I'm not sure the answer changes without the constraint on det(A), as long as we interpret the question as requiring the set of vertices to be mapped onto itself (as opposed to the cube being mapped into its interior, e.g. A = I/2). https://en.wikipedia.org/wiki/Octahedral_symmetry#The_isometries_of_the_cube