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Ma1bAnSol5 - magnification=1200 CALIFORNIA INSTITUTE OF...

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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 ) α m ( a, b ) , where α m ( a, b ) = m - 1 X i =0 a m - 1 - i b i . Further α 1 ( a, b ) = 1, α 2 ( a, b ) = a + b , and α 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 ) β, where β = 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 ) m ( a, b ), and ab m - ba m = ( b - a ) abα m - 1 ( a, b ) , so det( A ) = ( b - a ) γ m where γ m = c m - m ( a, b ) + abα m - 1 ( a, b ) . Next γ 2 = c 2 - c ( a + b ) + ab = ( c - a )( c - b ) , verifying (*) when m = 2. Similarly γ 3 = c 3 - c ( a 2 + ab + b 2 ) + ab ( a + b ) = ( c 2 - c ( a + b ) + ab )( a + b + c ) = ( c - a )( c - b )( a + b + c ) , verifying (*) when m = 3. Next let B = 1 1 1 a 2 b 2 c 2 a 3 b 3 c 3 . We use Theorem 3.9 to show (**) det( B ) = ( b - a )( c - a )( c - b )( ac + ab + bc ) .
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