document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006565>Question 1006565</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If A and B are symmetric matrices and if AB = BA, then AB is symmetric\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "     is this statement true or false and why? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622674 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=622674\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A matrix is <a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Symmetric_matrix\">symmetric</a> if and only if it is equal to its transpose, ie<BR>\n" );
document.write( "X = X^T\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Given:<BR>\n" );
document.write( "A = A^T (since matrix A is symmetric)<BR>\n" );
document.write( "B = B^T (matrix B is symmetric)<BR>\n" );
document.write( "AB = BA\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We want to prove:<BR>\n" );
document.write( "AB is symmetric<BR>\n" );
document.write( "ie, AB = (AB)^T\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "AB = BA<BR>\n" );
document.write( "AB = B^T*A^T ... use the given info above<BR>\n" );
document.write( "AB = (AB)^T ... use <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Transpose\">property 3</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the claim has been proven true. AB is indeed symmetric. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );