Let A and B be square matrices of order n, such that B = 2A - I. (1) One has to proove that B is involutory (B^2 = I) <=> A is idempotent (A^2 = A). Starting off with direction <= , one multiplies (1) with A from the right and from the left, separately, and obtains the results BA = A, and AB = A, respectively. This implies B = I, and I is an involutory matrix, so that direction is prooved. (I hope, that's why I'm asking.) The second direction, => , causes trouble and I'd appreciate any help. I tried multiplying (1) with B, but it didn't seem to help. Thanks in advance.