Final answer:
The matrix equation A³ = I in 3D can be solved by considering linear transformations, specifically rotations of 120° around an axis (C3 axes) and combinations of rotations and reflections that return to the original configuration when applied three times.
Step-by-step explanation:
The question is asking to find all possible solutions to the matrix equation A³ = I, where I is the identity matrix in 3D. To approach this problem, we can think of it in terms of linear transformations. A matrix A for which A³ = I represents a transformation in three-dimensional space that when applied three times successively, results in the original configuration.
One example of such a transformation is a rotation. In three dimensions, rotations that are multiples of 120° around an axis (like the diagonals of a cube, or C3 axes) result in the cube appearing unchanged after three such rotations. Hence, these rotations can represent solutions to our original matrix equation. To find matrices that correspond to these rotations, one can use Euler's rotation theorem to construct rotation matrices around specified axes by the correct angles.
If we want to think about this in terms of matrix properties, we are looking for matrices that are orthogonal (A^{-1} = A^{T}), since orthogonal matrices represent rotations and reflections that preserve the structure of the space. Specifically, we're interested in the subset that when cubed, result in the identity matrix. Thus, the solutions include not only rotation matrices of angles that are multiples of 120° but also combinations of rotations and possibly reflections that achieve the same overall transformation when applied three times.