MATH311ansHW9

MATH311ansHW9 - Math 311, Homework 9 partial answers and...

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Unformatted text preview: Math 311, Homework 9 partial answers and solutions 3.6C.2. Answer: Theorem 6.7 does not apply, no basis of eigenvectors. 3.6C.3. The characteristic polynomial det- 1- 1- = 2 + 1 has two distinct complex roots = i . Thus the matrix has a basis of eigenvectors in C 2 , but not in R 2 . 3.6C.8. The characteristic polynomial is det cos - - sin sin cos - = (cos - ) 2 + sin 2 = cos 2 - 2 cos + 2 + sin 2 = 2- 2 cos + 1 . Using the quadratic formula, its discriminant is D = 4(cos 2 - 1) =- 4 sin 2 , and so = cos i sin . The corresponding eigenvectors are in the null spaces of i sin - sin sin i sin , so they are ( i 1 ) . 3.6C.11. The characteristic polynomial is det 4- - 3 2- 1- =- 4 + - 4 + 2 + 6 = 2- 3 + 2 = ( - 1)( - 2) . Therefore the eigenvalues are = 1 , 2 . The eigenvectors corresponding to = 1 lie in the null space of 3- 3 2- 2 so one of them is ( 1 1 ) . The eigenvectors corresponding to....
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MATH311ansHW9 - Math 311, Homework 9 partial answers and...

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