document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1044576>Question 1044576</A>: <I><SPAN STYLE=\"word-break: break-all;\"> |a b c d 1|<BR>\n" );
document.write( "|b c d a 1|<BR>\n" );
document.write( "|c d a b 1|  =0<BR>\n" );
document.write( "|d a b c 1|<BR>\n" );
document.write( "|b a d c 1|\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #659894 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=659894\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%285%2C5%2Ca%2Cb%2Cc%2Cd%2C1%2Cb%2Cc%2Cd%2Ca%2C1%2Cc%2Cd%2Ca%2Cb%2C1%2Cd%2Ca%2Cb%2Cc%2C1%2Cb%2Ca%2Cd%2Cc%2C1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Perform the following elementary column operations (ECO3) IN SUCCESSION (which don't change the value of the determinant):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2AC%5B1%5D+%2B+C%5B4%5D ALIGN=MIDDLE  ALT=\"1%2AC%5B1%5D+%2B+C%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2AC%5B2%5D+%2B+C%5B4%5D ALIGN=MIDDLE  ALT=\"1%2AC%5B2%5D+%2B+C%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2AC%5B3%5D+%2B+C%5B4%5D ALIGN=MIDDLE  ALT=\"1%2AC%5B3%5D+%2B+C%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "What you get is the determinant\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%285%2C5%2Ca%2Cb%2Cc%2Ca%2Bb%2Bc%2Bd%2C1%2Cb%2Cc%2Cd%2Ca%2Bb%2Bc%2Bd%2C1%2Cc%2Cd%2Ca%2Ca%2Bb%2Bc%2Bd%2C1%2Cd%2Ca%2Cb%2Ca%2Bb%2Bc%2Bd%2C1%2Cb%2Ca%2Cd%2Ca%2Bb%2Bc%2Bd%2C1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%2Bc%2Bd%29abs%28matrix%285%2C5%2Ca%2Cb%2Cc%2C1%2C1%2Cb%2Cc%2Cd%2C1%2C1%2Cc%2Cd%2Ca%2C1%2C1%2Cd%2Ca%2Cb%2C1%2C1%2Cb%2Ca%2Cd%2C1%2C1%29%29=0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "since columns 4 and 5 have identical entries.  (Note that a+b+c+d was factored ONCE only from the fourth column.)<BR>\n" );
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