sample_tt2_2_ans

sample_tt2_2_ans - Math 235 Sample Term Test 2 - 2 Answers...

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Unformatted text preview: Math 235 Sample Term Test 2 - 2 Answers NOTE : - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) State the Principal Axis Theorem. Solution: A matrix is symmetric matrix if and only if it is orthogonally diagonalizable. b) Let A be an m n matrix. Prove that A T A is symmetric. Solution: ( A T A ) T = A T A TT = A T A , hence A T A is symmetric. c) State the definition of a quadratic form Q ( vectorx ) on R n being negative definite. Solution: Q ( vectorx ) is negative definite if Q ( vectorx ) < 0 for all vectorx negationslash = vector 0. d) Consider the quadratic form Q ( vectorx ) = 3 x 2 + 5 y 2 + 3 z 2- 2 xy- 2 xz + 2 yz . Write down the symmetric matrix A such that Q ( vectorx ) = vectorx T Avectorx . Solution: A = 3- 1- 1- 1 5 1- 1 1 3 . e) For the matrix A = - 7- 2- 3- 5 5- 1 1 2- 5- 5 6 4 2 7- 10- 4- 4- 5 10 10- 6- 2- 3- 5 7 2 6 2 3 5- 5 ,- 1 is an eigenvalue with multiplicity three, and 2 is an eigenvalue with multiplicity two. Determine the determinant of A . Solution: det A = (- 1)(- 1)(- 1)(2)(2)(- 3) = 12 . 2. Let A = 4 2 2 2 4 2 2 2 4 . Find an orthogonal matrix P that diagonalizes A and the corresponding diagonal matrix....
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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sample_tt2_2_ans - Math 235 Sample Term Test 2 - 2 Answers...

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