Question: The transformation y = AX where turns a point with coordinates (x 1 , x 2 , x 3 ) into a point with coordinates

The transformation y = AX where

A = 8-1-4 4 4 7 1 -8 4 Y y = y Y3 [x and X X = X [X3]

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.

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