document.write( "Question 350511: (a)If A is idempotent and has an inverse A^(-1) show that A=I
\n" ); document.write( "Explain what this result says about the kinds of matrices that could possibly be both invertible and idempotent.\r
\n" ); document.write( "\n" ); document.write( "(b) If A is idempotent, does it have to be square? Why or why not? \r
\n" ); document.write( "\n" ); document.write( "(c) Let B=[1 0
\n" ); document.write( " 0 0] Is B idempotent?
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Algebra.Com's Answer #250538 by jim_thompson5910(35256)\"\" \"About 
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a) If A is idempotent, then \"A%5E2=A\". You can break this down to say that \"A%2AA=A\". From there, left multiply both sides by \"A%5E%28-1%29\" (you can right multiply both sides by \"A%5E%28-1%29\" also) to get \"A%5E%28-1%29A%2AA=A%5E%28-1%29%2AA\" which then becomes \"I%2AA=I\" which simplifies to \"A=I\"\r
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\n" ); document.write( "\n" ); document.write( "b) The matrix A is idempotent when \"A%5E2=A\". In order for \"A%5E2=A\" to be true, \"A%5E2=A%2AA\" must be defined (ie possible). So in order for \"A%2AA\" to be defined, A must have the same number of rows and columns. This means that A must be square.\r
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\n" ); document.write( "\n" ); document.write( "c) Does \"B%5E2=B\" ? If so, then matrix B is idempotent. Notice how . So this shows us that \"B%2AB=B\" and that \"B%5E2=B\". So matrix B is idempotent. \n" ); document.write( "
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