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math102HW5

# math102HW5 - Math 102 Homework 5(selected problems David...

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Math 102 - Homework 5 (selected problems) David Lipshutz Problem 1. (Strang, 4.2: #2) If a 3 by 3 matrix has det A = - 1, find det( 1 2 A ), det( - A ), det( A 2 ), and det( A - 1 ). Proof. det( 1 2 A ) = det( 1 2 I ) det( A ) = ( 1 2 ) 3 ( - 1) = - 1 8 det( - A ) = det( - I ) det( A ) = ( - 1) 3 ( - 1) = 1 det( A 2 ) = det( A ) det( A ) = ( - 1)( - 1) = 1 det( A - 1 ) = det( A ) - 1 = - 1 Problem 2. (Strang, 4.2: #10) If Q is an orthogonal matrix, so that Q T Q = I , prove that det Q equals +1 or - 1. What kind of box is formed from the rows (or columns) of Q ? Proof. We have 1 = det I = det( Q T Q ) = det Q T det Q = (det Q ) 2 So det Q = ± 1. The box formed has volume 1 and orthogonal edges. Problem 3. (Strang, 4.2: #12) Use row operations to verify that the 3 by 3 ”Vandermonde determinant” is det 1 a a 2 1 b b 2 1 c c 2 = ( b - a )( c - a )( c - b ) Proof. det 1 a a 2 1 b b 2 1 c c 2 = det 1 a a 2 0 b - a b 2 - a 2 0 c - a c 2 - a 2 ( R 2 - R 1) = det 1 a a 2 0 b - a b 2 - a 2 0 0 c 2 - a 2 - ( c - a )( b + a ) ( R 3 - c - a b - a R 2) = det 1 a a 2 0 b - a b 2 - a 2 0 0 ( c - b )( c - a ) = ( b - a )( c

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math102HW5 - Math 102 Homework 5(selected problems David...

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