document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=190355>Question 190355</A>: <I><SPAN STYLE=\"word-break: break-all;\">  I need HELP using the Gauss-Jordan method to find A<sup>-1</sup> if it exists,\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C1%2C1%2C1%2C2%2C1%2C1%2C2%2C2%2C3%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C1%2C1%2C2%2C1%2C1%2C2%2C2%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142901 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=142901\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Simply append the 3x3 identity matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the given matrix to get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "  <BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C6%2C1%2C1%2C1%2C1%2C0%2C0%2C2%2C1%2C1%2C0%2C1%2C0%2C2%2C2%2C3%2C0%2C0%2C1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now use Gauss-Jordan Elimination (ie row reduce) to transform the left hand block matrix to the 3x3 identity matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The right hand block 3x3 matrix will be the inverse of the given matrix.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So here are the steps needed to row reduce (provided by the <a  rel=nofollow HREF=\"http://www.math.odu.edu/~bogacki/lat/\">Linear Algebra Toolkit</a>):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/rref-1.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how the left hand block matrix is  the 3x3 identity matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so this means that 1) the inverse of A exists (and is unique), and 2) the right hand matrix is the inverse of A\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the right hand block 3x3 matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C-1%2C1%2C0%2C4%2C-1%2C-1%2C-2%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C-1%2C1%2C0%2C4%2C-1%2C-1%2C-2%2C0%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C1%2C1%2C1%2C2%2C1%2C1%2C2%2C2%2C3%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C1%2C1%2C2%2C1%2C1%2C2%2C2%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E%28-1%29=%28matrix%283%2C3%2C-1%2C1%2C0%2C4%2C-1%2C-1%2C-2%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"A%5E%28-1%29=%28matrix%283%2C3%2C-1%2C1%2C0%2C4%2C-1%2C-1%2C-2%2C0%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );