document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1101996>Question 1101996</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me<BR>\n" );
document.write( "given that<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+ ALIGN=MIDDLE  ALT=\"+A+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is non-degenerate and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > is multiplication commutable with<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A ALIGN=MIDDLE  ALT=\"+A\"   DISABLED_style= \"max-width:90%; height:auto;\" > prove that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+ ALIGN=MIDDLE  ALT=\"B+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and <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;\" > also multiplication commutable </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #716672 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=716672\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "Given Fact #1: Matrix A is non-degenerate<BR>\n" );
document.write( "This means that the inverse of A exists. The inverse is denoted A^(-1)\r<BR>\n" );
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document.write( "Given Fact #2: Matrix B is multiplication commutable with matrix A.<BR>\n" );
document.write( "In other words, A*B = B*A holds true. The order of multiplication doesn't matter in this specific case (keep in mind that matrix multiplication isn't always commutative)\r<BR>\n" );
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document.write( "The goal is to prove that matrix B and the inverse A^(-1) are also multiplication commutable. We need to show the following:<BR>\n" );
document.write( "A^(-1)*B = B*A^(-1)<BR>\n" );
document.write( "or<BR>\n" );
document.write( "B*A^(-1) = A^(-1)*B\r<BR>\n" );
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document.write( "Here's one way to do that\r<BR>\n" );
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document.write( "A*B = B*A <font color=blue>Start with fact #2</font><BR>\n" );
document.write( "A^(-1)*A*B = A^(-1)*B*A <font color=blue>Left-Multiply both sides by A^(-1). This step is possible because of fact #1</font><BR>\n" );
document.write( "I*B = A^(-1)*B*A<BR>\n" );
document.write( "B = A^(-1)*B*A<BR>\n" );
document.write( "B*A^(-1) = A^(-1)*B*A*A^(-1)<font color=blue>Right-Multiply both sides by A^(-1)</font><BR>\n" );
document.write( "B*A^(-1) = A^(-1)*B*I<BR>\n" );
document.write( "B*A^(-1) = A^(-1)*B\r<BR>\n" );
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document.write( "So that shows A^(-1) and B are multiplication commutable\r<BR>\n" );
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document.write( "Here's another way\r<BR>\n" );
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document.write( "A*B = B*A<BR>\n" );
document.write( "A*B*A^(-1) = B*A*A^(-1) <font color=blue>Right-Multiply both sides by A^(-1)</font><BR>\n" );
document.write( "A*B*A^(-1) = B*I<BR>\n" );
document.write( "A*B*A^(-1) = B<BR>\n" );
document.write( "B = A*B*A^(-1)<BR>\n" );
document.write( "A^(-1)*B = A^(-1)*A*B*A^(-1)<font color=blue>Left-Multiply both sides by A^(-1)</font><BR>\n" );
document.write( "A^(-1)*B = I*B*A^(-1)<BR>\n" );
document.write( "A^(-1)*B = B*A^(-1)\r<BR>\n" );
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document.write( "Giving us the same conclusion as before. Keep in mind that if X = Y, then Y = X for any matrices X and Y. <BR>\n" );
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