document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=243802>Question 243802</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would you find the determinant of the n x n matrix:<BR>\n" );
document.write( "a b 0 0...0<BR>\n" );
document.write( "0 a b 0...0<BR>\n" );
document.write( "0 0 a b...0<BR>\n" );
document.write( "...........<BR>\n" );
document.write( "0 0 0 0...b<BR>\n" );
document.write( "b 0 0 0...a\r<BR>\n" );
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document.write( "I understand that i need to use laplace expansion but this problem is confusing me just because you dont know the numbers in the matrix or exactly how many rows and columns you are working with </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178592 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=178592\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we look at the 2x2 matrix\r<BR>\n" );
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document.write( "we can clearly see that the determinant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just use the definition of the 2x2 determinant).\r<BR>\n" );
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document.write( "Moving onto the 3x3 matrix\r<BR>\n" );
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document.write( "a b 0<BR>\n" );
document.write( "0 a b<BR>\n" );
document.write( "b 0 a\r<BR>\n" );
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document.write( "If we compute this determinant, we get\r<BR>\n" );
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document.write( "Similarly, for the 4x4 matrix\r<BR>\n" );
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document.write( "a b 0 0 <BR>\n" );
document.write( "0 a b 0 <BR>\n" );
document.write( "0 0 a b <BR>\n" );
document.write( "b 0 0 a \r<BR>\n" );
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document.write( "If we apply the laplace expansion along the first column, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Aabs%28matrix%283%2C3%2Ca%2Cb%2C0%2C0%2Ca%2Cb%2C0%2C0%2Ca%29%29-b%2Aabs%28matrix%283%2C3%2Cb%2C0%2C0%2Ca%2Cb%2C0%2C0%2Ca%2Cb%29%29=a%2Aa%5E3-b%2Ab%5E3=a%5E4-b%5E4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >. Also, for the 5x5 matrix\r<BR>\n" );
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document.write( "a b 0 0 0<BR>\n" );
document.write( "0 a b 0 0<BR>\n" );
document.write( "0 0 a b 0<BR>\n" );
document.write( "0 0 0 a b<BR>\n" );
document.write( "b 0 0 0 a\r<BR>\n" );
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document.write( "the determinant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E5%2Bb%5E5 ALIGN=MIDDLE  ALT=\"a%5E5%2Bb%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Are you starting to see a pattern? If not, then try to find the determinant of the 6x6 matrix \r<BR>\n" );
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document.write( "a b 0 0 0 0<BR>\n" );
document.write( "0 a b 0 0 0<BR>\n" );
document.write( "0 0 a b 0 0<BR>\n" );
document.write( "0 0 0 a b 0<BR>\n" );
document.write( "0 0 0 0 a b<BR>\n" );
document.write( "b 0 0 0 0 a\r<BR>\n" );
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document.write( "You'll find that the determinant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E6-b%5E6 ALIGN=MIDDLE  ALT=\"a%5E6-b%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In general, the determinant of the n x n matrix\r<BR>\n" );
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document.write( "a b 0 0...0<BR>\n" );
document.write( "0 a b 0...0<BR>\n" );
document.write( "0 0 a b...0<BR>\n" );
document.write( "...........<BR>\n" );
document.write( "0 0 0 0...b<BR>\n" );
document.write( "b 0 0 0...a\r<BR>\n" );
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document.write( "is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5En%2B%28-1%29%5E%28n%2B1%29b%5En ALIGN=MIDDLE  ALT=\"a%5En%2B%28-1%29%5E%28n%2B1%29b%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I left some work out due to time constraints. Let me know if you need to see how to get any determinants. </SPAN>\n" );
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