document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=475047>Question 475047</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am reading my solutions and I don't understand how I get from one tableau to another. We pivoted around the second row and second column.<BR>\n" );
document.write( "1   2  4  1  0  0 |  8<BR>\n" );
document.write( "3   8  1  0  1  0 |  10<BR>\n" );
document.write( "-4 -32 1  0  0  1 |  0\r<BR>\n" );
document.write( "\n" );
document.write( "to \r<BR>\n" );
document.write( "\n" );
document.write( "1  0  15  4  -1  0  | 22<BR>\n" );
document.write( "3  8  1   0  1   0  | 10<BR>\n" );
document.write( "8  0  5   0  4   1  |  40 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #325737 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=325737\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Any row in a matrix, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large R_i\ =\ \alpha_{i,1}\ \alpha_{i,2}\ \cdots\ \alpha_{i,j}\ \cdots\ \alpha_{i,n\,-\,1}\ \alpha_{i,n}\"> can be replaced by the sum of a constant times any other row plus row <i>i</i>.  That is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\,\cdot\,R_j\ +\ R_i\ \Rightarrow\ \ R_i\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Actually, in more general terms, you can multiply both rows by constants if it makes sense to do so, thus:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\,\cdot\,R_j\ +\  b\,\cdot\,R_i\ \Rightarrow\ \ R_i\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1.  Multiply Row 2 by 4:  12 32 4 0 4 0 | 40\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2.  Add the result of Step 1 to Row 3:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "12 + (-4) = 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "32 + (-32) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4 + 1 = 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "0 + 0 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4 + 0 = 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "0 + 1 = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "40 + 0 = 40\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hence: 8 0 5 0 0 4 1 | 40\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3.  Replace Row 3 with the results of Step 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: Just because you multiplied Row 2 by 4 in order to drive the 3rd Row 2nd Column element to zero, doesn't mean you have to replace Row 2 with the results of the multiplication.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );