document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=3715>Question 3715</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determinant<BR>\n" );
document.write( "show that determinant<BR>\n" );
document.write( "b+c,a,a,<BR>\n" );
document.write( "b,c+a,b<BR>\n" );
document.write( "c.c. a+b<BR>\n" );
document.write( "=4abc<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #1648 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=1648\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  |b+c a a|<BR>\n" );
document.write( " |b c+a b|<BR>\n" );
document.write( " |c c a+b|\r<BR>\n" );
document.write( "\n" );
document.write( " = (a+b)(b+c)(c+a) + 2abc - ca(c+a) - bc(b+c) - ab(a+b)<BR>\n" );
document.write( " = (b^2+ab+ca+bc)(c+a) + 2abc - ca(c+a) - bc(b+c) - ab(a+b)<BR>\n" );
document.write( " =  b^2c+abc+c^2a+bc^2 + ab^2+a^2b+ca^2+abc + 2abc - ca(c+a) - bc(b+c) - ab(a+b)<BR>\n" );
document.write( " =  (b^2c+ bc^2)+(c^2a+ ca^2) +(a^2b+ ab^2)+42abc - ca(c+a) - bc(b+c) - ab(a+b)<BR>\n" );
document.write( " = 4abc.\r<BR>\n" );
document.write( "\n" );
document.write( " Or let this determinant be f(a,b,c).\r<BR>\n" );
document.write( "\n" );
document.write( " We see that f(0,b,c) = (b+c) (cb - bc) = 0<BR>\n" );
document.write( " Similarly, f(a,0,c) = f(a,b,0) = 0<BR>\n" );
document.write( " <BR>\n" );
document.write( " Hence,abc is a divisor of f(a,b,c).<BR>\n" );
document.write( " Since deg f(a,b,c) = 3, assume f(a,b,c) = k abc.\r<BR>\n" );
document.write( "\n" );
document.write( " Set a=1,b=1,c =-1, we get<BR>\n" );
document.write( " |0  1  1|<BR>\n" );
document.write( " |1  0  1| = -1+(-1)-2 = -4 = k*1*1*(-1)= -k<BR>\n" );
document.write( " |-1 -1 2|\r<BR>\n" );
document.write( "\n" );
document.write( " So, k = 4 and f(a,b,c) = 4 abc.\r<BR>\n" );
document.write( "\n" );
document.write( " I prefer the 2nd method and hope that you can understand it.<BR>\n" );
document.write( " Kenny\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );