Question

If A is idempotent, show that I-2A is equal to I-2A inverse.

Answer 1

Consider \[(I-2A)^{2} = I-4A+4A^{2}\]
Because A is idempotent, \[A=A^{2}\]
so our equation becomes\[I-4A+4A = I\]
If something squared is the identity, it is its own inverse.