document.write( "Question 1067883: How to proofs A is idempotente if and only if (I-A) is idempotente \n" ); document.write( "

Algebra.Com's Answer #683111 by ikleyn(52908) $\"\"$ $\"About$

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\n" ); document.write( "How to $\"highlight%28cross%28proofs%29%29\"$ prove A is $\"highlight%28cross%28idempotente%29%29\"$ idempotent if and only if (I-A) is $\"highlight%28cross%28idempotente%29%29\"$ idempotent.
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\n" ); document.write( "\n" ); document.write( "Definition (see Wikipedia, https://en.wikipedia.org/wiki/Idempotent_matrix )\r
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document.write( "    In algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. \r\n" );
document.write( "    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.\r\n" );
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document.write( "1.  Let A be an idempotent. Then A*A = A, by the definition.\r\n" );
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document.write( "    Then (1-A)*(1-A) = 1 - A - A + (-A)*(-A) = 1 - 2A + A*A = 1 - 2A + A = 1-A.   \r\n" );
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document.write( "                                                      ( I replaced A*A by A using the fact (the assumption) that A is idempotent ).\r\n" );
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document.write( "    Thus we proved that \"If A is idempotent, THEN (1-A) is idempotent.\"\r\n" );
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document.write( "2.  Let 1-A be an idempotent.  Let denote  B = 1-A  then.\r\n" );
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document.write( "    According to #1,  then 1-B is idempotent.\r\n" );
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document.write( "                      But  1-B = 1 - (1-A) = A.\r\n" );
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document.write( "                      Hence, A is idempotent.\r\n" );
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document.write( "    Thus we proved that \"If 1-A is idempotent, THEN A is idempotent.\"\r\n" );
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document.write( "3.  So, we proved the statement \"A is idempotent if and only if (I-A) is idempotent.\"\r\n" );
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