math102HW5

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

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Unformatted text preview: 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 b- a b 2- a 2 c- a c 2...
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This note was uploaded on 04/03/2011 for the course MATH 102 taught by Professor Szypowsk during the Fall '08 term at UCSD.

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

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