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340finalexamsolfall07

# 340finalexamsolfall07 - MATH 340 FINAL EXAM 150 points...

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MATH 340; FINAL EXAM, 150 points, December 17 , 2007 (R.A.Brualdi) TOTAL SCORE : Name: These R. Solutions I. (24 points; 3 points each) Answer the following short answer questions (no justiFcation wanted). If the information given is not enough to uniquely determine the answer, write undetermined . Let A be an n by n matrix with eigenvalues (including multiplicities) 3, 3, 4, 4, 4. 1. The order n of A is: 5 2. The determinant of A is: 3 2 4 3 = 576 3. The coe±cient of λ 4 in the characteristic polynomial of A is: - (3+3+4+4+4) = - 18 4. The dimension of the row space of A is: 5 5. The eigenvalues of the matrix A 2 of A are:3 2 , 3 2 , 4 2 , 4 2 , 4 2 . 6. Is A invertible? YES (no eigenvalue is 0) 7. The dimension of the eigenspace of A for the eigenvalue 3 is: Undetermined 8. Is A diagonalizable? Undetermined II. (12 points; 2 points each) Circle whether the following assertions are T rue or F alse: 1. ²: A real, square matrix always has at least one real eigenvalue. 2. T: A Fnite dimensional vector space with an inner product always has an orthonormal basis. 3. T : Every real, symmetric matrix is diagonalizable. . 4. T: If

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340finalexamsolfall07 - MATH 340 FINAL EXAM 150 points...

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