a) Suppose the product B¹ B is invertible for some matrix B € Rmxk. Show that B (BTB)¯ is idempotent. b) If A is idempotent, show that Inxn – A is also. c) If A is idempotent, show that Inxn - A is invertible by giving an explicit formula for its inverse. d) Suppose that A is idempotent and that we are given x 0 and X satisfying Ax = Xx. Show that λ = {0, 1}.

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