document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=102267>Question 102267</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Multiplication of matrices are only commutative for some cases of matrices. What are these cases? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #74315 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=74315\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Off the top of my head I can only think of these cases where commutativity holds:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I%2AA=A%2AI ALIGN=MIDDLE  ALT=\"I%2AA=A%2AI\"   DISABLED_style= \"max-width:90%; height:auto;\" > where I is the identity matrix\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E-1%2AA=A%2AA%5E-1 ALIGN=MIDDLE  ALT=\"A%5E-1%2AA=A%2AA%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E-1 ALIGN=MIDDLE  ALT=\"A%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the inverse of matrix A\r<BR>\n" );
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document.write( "Also, according to <a  rel=nofollow HREF=http://mathworld.wolfram.com/MatrixMultiplication.html>Wolfram Mathworld</a>, \"...matrix multiplication is not, in general, commutative (although it is commutative  if A and B are diagonal and of the same dimension).\"\r<BR>\n" );
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document.write( "For instance, if you have matrices diagonal matrices A and B\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C2%2C0%2C0%2C0%2C3%2C0%2C0%2C0%2C5%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C2%2C0%2C0%2C0%2C3%2C0%2C0%2C0%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=%28matrix%283%2C3%2C5%2C0%2C0%2C0%2C6%2C0%2C0%2C0%2C7%29%29 ALIGN=MIDDLE  ALT=\"B=%28matrix%283%2C3%2C5%2C0%2C0%2C0%2C6%2C0%2C0%2C0%2C7%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the first product AB is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=%28matrix%283%2C3%2C10%2C0%2C0%2C0%2C18%2C0%2C0%2C0%2C35%29%29 ALIGN=MIDDLE  ALT=\"AB=%28matrix%283%2C3%2C10%2C0%2C0%2C0%2C18%2C0%2C0%2C0%2C35%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the second product BA is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BA=%28matrix%283%2C3%2C10%2C0%2C0%2C0%2C18%2C0%2C0%2C0%2C35%29%29 ALIGN=MIDDLE  ALT=\"BA=%28matrix%283%2C3%2C10%2C0%2C0%2C0%2C18%2C0%2C0%2C0%2C35%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which is the same product as AB. So if you were to do this with general entries of the matrices A and B, you would find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=BA ALIGN=MIDDLE  ALT=\"AB=BA\"   DISABLED_style= \"max-width:90%; height:auto;\" > only if A and B are diagonal matrices and they are both the same size </SPAN>\n" );
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