Ma1bAnSol5

Ma1bAnSol5 - magnification=1200 CALIFORNIA INSTITUTE OF...

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Unformatted text preview: magnification=1200 CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b; Solutions to Homework Set 5 Due: February 19, 2008 1. Observe first that for a,b R and m a positive integer, (!) a m- b m = ( a- b ) m ( a,b ) , where m ( a,b ) = m- 1 X i =0 a m- 1- i b i . Further 1 ( a,b ) = 1, 2 ( a,b ) = a + b , and 3 ( a,b ) = a 2 + ab + b 2 . Let a,b,c R and A = 1 1 1 a b c a m b m c m . We will use Theorem 3.9 to show (*) det( A ) = ( b- a )( c- a )( c- b ) , where = 1 ,a + b + c for m = 2 , 3, respectively. By Theorem 3.9, det( A ) = det( A 1 , 1 )- det( A 1 , 2 ) + det( A 1 , 3 ) = ( bc m- cb m )- ( ac m- ca m ) + ab m- ba m . Now bc m- ac m = ( b- a ) c m , cb m- ca m = ( b- a ) c m ( a,b ), and ab m- ba m = ( b- a ) ab m- 1 ( a,b ) , so det( A ) = ( b- a ) m where m = c m- c m ( a,b ) + ab m- 1 ( a,b ) . Next 2 = c 2- c ( a + b ) + ab = ( c- a )( c- b ) , verifying (*) when m = 2. Similarly 3 = c 3- c ( a 2...
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This note was uploaded on 04/28/2010 for the course MATH 1B taught by Professor Aschbacher during the Winter '07 term at Caltech.

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Ma1bAnSol5 - magnification=1200 CALIFORNIA INSTITUTE OF...

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