document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=852815>Question 852815</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello! \r<BR>\n" );
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document.write( "I need a lot of help here. I have a question that need to be proved but I don't know how to do.\r<BR>\n" );
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document.write( "A=([1,0,0,0,0],[a,-2,0,0,0],[a^2,b,-3,0,0],[a^3,b^2,c,-4,0],[a^4,b^3,c^2,d,-5])\r<BR>\n" );
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document.write( "there are four answers provided.<BR>\n" );
document.write( "a)|A|=5!<BR>\n" );
document.write( "b)|A|=(-1)^5(1.2.3.4.5)<BR>\n" );
document.write( "c)|A|=0 since the rows are related<BR>\n" );
document.write( "d)It is not possible to calculate the determinant of A because we don't know the value of a,b,c and d.\r<BR>\n" );
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document.write( "Thank you! Your help means a world to me!   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #513692 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=513692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> From this page, <a  rel=nofollow HREF=\"http://www.math.lsa.umich.edu/~hochster/419/det.html\">http://www.math.lsa.umich.edu/~hochster/419/det.html</a>, it states that \r<BR>\n" );
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document.write( "<blockquote><font color=\"blue\">Fact 7. The determinant of a lower triangular matrix (or an upper triangular matrix) is the product of the diagonal entries. In particular, the determinant of a diagonal matrix is the product of the diagonal entries.</font></blockquote> \r<BR>\n" );
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document.write( "Since matrix A=([1,0,0,0,0],[a,-2,0,0,0],[a^2,b,-3,0,0],[a^3,b^2,c,-4,0],[a^4,b^3,c^2,d,-5]), which really looks like this\r<BR>\n" );
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document.write( "[1,0,0,0,0]<BR>\n" );
document.write( "[a,-2,0,0,0]<BR>\n" );
document.write( "[a^2,b,-3,0,0]<BR>\n" );
document.write( "[a^3,b^2,c,-4,0]<BR>\n" );
document.write( "[a^4,b^3,c^2,d,-5]\r<BR>\n" );
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document.write( "is <a  rel=nofollow HREF=\"http://mathworld.wolfram.com/LowerTriangularMatrix.html\">lower triangular</a>, this means that we just multiply the diagonal entries, which are: 1, -2, -3, -4, -5\r<BR>\n" );
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document.write( "So multiplying them gives us: 1*(-2)*(-3)*(-4)*(-5) = 120\r<BR>\n" );
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document.write( "which is equivalent to 5! since 5! = 5*4*3*2*1 = 120\r<BR>\n" );
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document.write( "The answer is <font size=4 color=\"red\">a)|A|=5!</font> </SPAN>\n" );
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