The transformation y = AX where

turns a point with coordinates (x₁, x₂, x₃) into a point with coordinates ( y₁, y₂, y₃). Show that the coordinates of the points that transform into themselves satisfy the matrix equation BX = 0, where B = A – I, with I the identity matrix. Find the rank of B and hence deduce that for points which transform into themselves

where α is a parameter. Find AA^T. What is the inverse of A? If y₁ = 3, y₂ = –1 and y₃ = 2, determine the values of x₁, x₂ and x₃ under this transformation.