SOLUTION: How to proofs A is idempotente if and only if (I-A) is idempotente

    In algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. 
    That is, the matrix M is idempotent if and only if MM = M. For this product MM to be defined, M must necessarily be a square matrix.

1.  Let A be an idempotent. Then A*A = A, by the definition.

    Then (1-A)*(1-A) = 1 - A - A + (-A)*(-A) = 1 - 2A + A*A = 1 - 2A + A = 1-A.   
 
                                                      ( I replaced A*A by A using the fact (the assumption) that A is idempotent ).

    Thus we proved that "If A is idempotent, THEN (1-A) is idempotent."


2.  Let 1-A be an idempotent.  Let denote  B = 1-A  then.

    According to #1,  then 1-B is idempotent.

                      But  1-B = 1 - (1-A) = A.

                      Hence, A is idempotent.

    Thus we proved that "If 1-A is idempotent, THEN A is idempotent."


3.  So, we proved the statement "A is idempotent if and only if (I-A) is idempotent."