document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=25058>Question 25058</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello:\r<BR>\n" );
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document.write( "I am not sure how to answer this question.\r<BR>\n" );
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document.write( "If A and B are nxn matrices, then show that there are no square matrices A and B such that AB-BA=I<BR>\n" );
document.write( "(where I is the Identity).\r<BR>\n" );
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document.write( "I know that if AB = I and BA = I then AB-BA cannot equal I.  Sorry I am confused.\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #13443 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=13443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "This is quite easy to prove by considering the trace of both sides. The Trace of a square matrix is simply the sum of it's diagonal elements. Hopefully you can see that if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?P=Q\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?Tr(P)=Tr(Q)\">, and that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?Tr(P+Q)=Tr(P)+Tr(Q)\">. If you can't see where these identities come from then please email back.\r<BR>\n" );
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document.write( "Ok, The matrix <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?AB\"> is given by the formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?AB_{ij}=\sum_k^n A_{ik}B_{kj}\">\r<BR>\n" );
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document.write( "and the trace of a matrix <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?P\"> is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sum_t^n P_{tt}\">\r<BR>\n" );
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document.write( "So the trace of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?AB\"> is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sum_t^n \sum_k^n A_{tk}B_{kt}\">\r<BR>\n" );
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document.write( "Doing a similar calculation for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?Tr(BA)\"> we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sum_t^n \sum_k^n B_{tk}A{kt}\">\r<BR>\n" );
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document.write( "Now you can either acccept that these are the same expression ie <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?Tr(AB)=Tr(BA)\"> or you can read my discussion in green of why we're assuming it is.\r<BR>\n" );
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document.write( "<font color=\"green\">Now you've not told me what the elements of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?A\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?B\"> are, if they were matrices(making A a matrix of matrices) then we would be in trouble, because we wouldn't have commutivity, but I'm gonna assume that we are working with some nice subfield of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\mathbb{C}\"> hence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?A_{ab}B_{cd}=B_{cd}A_ab}\">. I'm also assuming our matrices are finite which allows us to change the order of summation(else we need to worry about absolute convergence on compact subsets - what fun!) So now you are happy that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?Tr(AB)=Tr(BA)\"></font>\r<BR>\n" );
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document.write( "Well this is it, we're done. The trace of the LHS must be zero, and the trace of the RHS (the trace of the identity) definitly isn't zero, it's <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?n\">. So unless <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?n=0\"> (doesn't make sense) there are no matrices that can satisfy this.\r<BR>\n" );
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document.write( "Hope that helps.\r<BR>\n" );
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document.write( "Kev </SPAN>\n" );
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