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Download HELM Workbook 22 (Eigenvalues and Eigenvectors) EVS Questions
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What is the determinant of 9 11 17 19 0% 19 0% 17 0% 11 0% 9 1. 2. 3. 4. 7 3 1 2 What is the determinant of 0 28 44 -28 0% -2 8 0% 44 0% 28 0% 0 1. 2. 3. 4. 6 2 0 1 4 2 2 3 0 Which matrix represents the following system of equations? 1. 1 4 1 7 1 11 0% 0% 0% 3 3. 1 0 4 0 1 7 2 1 1 2. x=4 y=7 What is the solution to the following system of equations? x 1 + x2 = 3 2x1 - 6x2 = -10 s. .. e e er e ar ar Th 0% no an x2 =½ Th an d =1 0% i.. . 0% er e x2 =2 0% an d x2 =5 =1 x1 x1 =4 an =9/ 8 0% an d d x2 ... 0% x1 x1=-9/8 and x2=-30/8 x1=4 and x2=5 x1=1 and x2=2 x1=1 and x2=½ There are an infinite number of solutions 6. There are no solutions x1 1. 2. 3. 4. 5. What is the solution to the following system of equations? 6x1 + 9x2 = 3 2x1 + 3x2 = 5 an d =1 x1 0% 0% 0% 0% x2 ==1. 5/ .. 2 an d Th x2 er ... e ar e Th an er i.. e . ar e no s. .. /.. . 0% x2 =1 an d =0 x1 =2 an d x2 =0 0% x1 x1=2 and x2=0 x1=0 and x2=1/3 x1=1 and x2=-1/3 x1=-5/2 and x2=2 There are an infinite number of solutions 6. There are no solutions x1 1. 2. 3. 4. 5. Which of the following statements are true? (A) sum of eigenvalues = sum diagonal elements (trace) (B) product of eigenvalues = determinant of square matrix A (C) distinct eigenvalues = linearly dependent eigenvectors 0% (A ), (B )a nd (.. . (.. . 0% nd (.. . (B )a nd B B 0% ot h (A )a nd (.. . 0% ot h (A )a nl y ot h O B nl y O 0% (C ) 0% (B ) (A ) 0% nl y Only (A) Only (B) Only (C) Both (A) and (B) Both (A) and (C) Both (B) and (C) (A), (B) and (C) O 1. 2. 3. 4. 5. 6. 7. Which of the following statements is true? 1. -1 A has an eigenvalue 1 2. (A - k) has an eigenvalue 3. -1 (A - kI) has an eigenvalue 1 4. 0% 4 0% 3 0% 2 0% 1 None of the above are true Dominant eigenvalue = eigenvalue with largest magnitude 1. True 2. False 3. Don’t Know 0% ’t Kn o ls e w 0% D on Fa Tr ue 0% What is the characteristic equation of the following matrix? 6 2 4 1 1. 2. 3. 4. λ² - 7λ + 6 = 0 λ² - 7λ + 14 = 0 λ² - 7λ - 2 = 0 None of the above What are the eigenvalues of 4 0 0 2 2 and 4 0 and 4 0 and 2 6 and 8 6 an d 8 0% 2 an d 0 an d 0 an d 0% 4 0% 4 0% 2 1. 2. 3. 4. What are the eigenvalues of the following matrix? 8 4 2 2 1. 2. 3. 4. λ = 2, 4 λ = 2, 6 λ = 2, 8 λ = 4, 8 What are the eigenvalues for the following matrix? 0 5 5 5 5 0 5 0 5 1. 2. 3. 4. λ = -5, 0, 5 λ = -5, 5, 10 λ = 0, 5, 5 λ = 5, 5, 10 What are the eigenvalues for the following matrix? 3 0 1 0 3 0 15 5 1 1. 2. 3. 4. λ = -2, 3, 4 λ = -6, -2, 3 λ = -2, 3, 6 λ = -6, -3, 2 Matrix A given below has eigenvalues λ = 2, 4, 6. Without further calculation -1 write down the eigenvalues for A . 3 1 1 A 1 3 1 0 0 0 1. 1, 2, 3 0% 0% 4 0% 3 0% 2 4. 1 1 1 , , 2 4 6 1 3. 1 2 , ,1 3 3 2. 1 ,1,1 3 Which of the vectors below is an eigenvector, corresponding to the eigenvalue λ= 7 of the matrix 3 2 4 5 1. 4. 0% 0% 0% 0% 4 1 2 3 2 1 1 2 2 3. 2. 1 2 1 Which of the vectors below is an eigenvector, corresponding to the eigenvalue λ= 3 of the matrix 1 4 2 1 0 3 1 2 4 1. 1 2 0 0% 0% 0% 0% 4 4. 3 1 2 0 2 1 0 2 3. 2. 1 2 1 0 What are the eigenvectors of the following matrix? 5 5 1 1 1. 1 1 , 1 5 0% 0% 4 0% 3 0% 2 4. 1 5 , 1 1 1 3. 1 1 , 1 5 2. 1 1 , 1 5 Which of the following shows the eigenvectors for matrix A? 4. 0 3 1 1, 1, 1 0 0 0 0% 0% 0% 0% 4 0 3 1 1, 1 , 1 0 0 0 0 3 1, 1 0 0 3 3. 2. 2 0 3 1, 1 0 0 1 1. 2 0 1 A 1 1 0 0 0 2 Which set of vectors is linearly independent? 0% se th e ot h B on e of of th e ,2 ), (3 , ab ... ... 0% 4) 0% (1 ,6 ), (3 , 18 ) 0% N (1,6), (3,18) (1,2), (3,4) None of the above Both of the sets (1 1. 2. 3. 4. 1 2 3 1 3 4. 2 6 4 2 2 2 0% 0% 0% 0% 4 3. 3 8 1 2 1 2 2 3 3 2 2 1 2 2. 2 1. Normalise the eigenvector X. 1 3 X 2 1 Diagonalization means which of the following? 0% 0% ab ... a. .. th e of e on sf or m in g N M ul tip ly in g th . .. di a. th e in g dd A 0% .. 0% Tr an 1. Adding the diagonal elements of a matrix. 2. Multiplying the diagonal elements of a matrix. 3. Transforming a non-diagonal matrix. 4. None of the above. Why might we want to diagonalize a matrix? ... e th es th e fin d of e on N ot h of to sy Ea 0% ... 0% se e. w e. .. po tin g om pu C 0% .. 0% B 1. Computing powers of the matrix becomes easy. 2. Easy to find eigenvalues of a diagonal matrix. 3. Both of these reasons. 4. None of these reasons You can always diagonalize an n x n matrix with n distinct eigenvalues. 1. True 2. False 3. Don’t Know 0% ’t Kn o ls e w 0% D on Fa Tr ue 0% Below are eigenvectors of four 2x2 matrices. Which matrix is definitely diagonalizable? 1. 0 0 , 1 3 2. 1 0 , 0 3 3. 1 1 , 3 3 0% 4 0% 3 0% 2 0% 1 4. 1 3 , 1 3 0% 0% 0% 0% 4 4. 1 4 1 1 3 3. 3 4 1 1 2. 1 0 1 1 2 1. 1 0 1 1 Obtain the modal matrix P. 1 1 0 A . 1 2 The matrix A= 1 3 0 2 has eigenvalues 1 and 2 with respective eigenvectors - 4. 1 0 0 2 0% 0% 0% 0% 4 3. 1 4 0 2 calculate P AP1. 3 2. 1 2 0 2 1 1 2 1. 1 4 1 2 1 1 1 If 1 1 P1 0 1 1 0 and 2 3 A 4 5 What is A²? 1. 4. 0% 0% 0% 0% 4 13 23 23 41 3 16 21 28 37 4 9 16 25 2 3. 2. 1 4 6 8 10 2 3 A 4 5 5 What is A ? 1. 10 15 20 25 2. 32 243 1024 3125 3. 6140 8097 10796 14237 0% 4 0% 3 0% 2 10796 14237 6140 80972 0% 1 4. The eigenvalues of a symmetric matrix with real elements are... 1. Always complex 2. Always real 3. Either complex or real rc th e Ei A lw ay s om pl ex re a m pl ex co s ay lw A 0% ... 0% l 0% Which of the following is a symmetric matrix? 1. 1 5 2 5 3 7 2 7 2 0% 0% 0% 4 0% 3 4. 4 6 1 3 2 1 2 3 2 3 7 3 7 4 1 3. 2. 1 4 4 1 A square matrix A is said to be orthogonal if A A -1 T 1. True 2. False 3. Don’t Know no w 0% Do n’ tK se 0% Fa l Tr ue 0% Two n x 1 column vectors X and Y are orthogonal if XY=0 1. True 2. False 3. Don’t Know 0% ’t Kn o ls e w 0% D on Fa Tr ue 0% The eigenvalues of a symmetric matrix A are λ=0 and λ=10 9 3 A 3 1 X and Y are the eigenvectors for λ=0 and λ=10 respectively. Are X and Y orthogonal? 1. Yes 2. No 3. Don’t Know no w 0% Do n’ tK 0% No Ye s 0% An Hermitian matrix is one satisfying A A T 1. True 2. False 3. Don’t Know no w 0% Do n’ tK se 0% Fa l Tr ue 0% Is the following matrix Hermitian? 3 2i 3 i A 2i 0 1 3 i 1 5 1. Yes 2. No 3. Don’t Know 0% w o 0% D on ’t Kn o N Ye s 0% Separating the variables in s 2 s gives 1. s(t ) Ae2t 2. s(t ) 2 Ae2 s 3. s(t ) 2 Ae2t 0% 0% 4 0% 3 0% 2 s(t ) Ae 2 t 1 4. Write in matrix form the pair of coupled differential equations 2. x 2 5 x y 3 1 y 0% 0% 4 0% 3 0% 2 3. x 2 3 x y 5 1 y 4. x 2 3 x y 5 1 y 1 1. x 2 5 x y 3 1 y x 2 x 3 y y 5 x y Find the solution of the coupled differential equations x x 4 y y 3 y with initial conditions x(0)=1 and y(0)=3 1. x(t ) 2e t 3e3t y(t ) 3e3t 2. x(t ) 2e t 3e3t 3. y(t ) 3e3t x(t ) 2et 3et 4. 0% 0% 0% 4 0% 3 y (t ) 3e t 2 x(t ) 2et 3et 1 y (t ) 3et Given r 1r r . What is the general solution to a system of 2nd order differential equations for the negative eigenvalues 1 , 2 ? 2 1 1. r (K L)cos t 1 s (M N)sin 2 t 2. r Kcos1t Lsin 1t s Mcos2 t Nsin 2 t 3. r Kcos t Lsin t 1 2 s Mcos1t Nsin 2 t 1 s M(cos2 t sin 2 t) 0% 0% 0% 4 0% 1 3 1 2 4. r K(cos t sin t) An elastic membrane in the x1 x2 plane 2 2 with boundary circle x1 x2 1 is shown below. The membrane is stretched so the point P:(x1 , x2 ) goes over the point Q:( y1 , y2 ) where y1 7 4 x1 y Ax x y 4 7 2 2 Find the amount that the principle directions are stretched by ’t K no D on 7 ct or s fa y 0% w .. a. .. 0% a. ct or s 4 a. fa y B B y fa ct or s 7 a. 3 ct or s fa y 0% .. 0% .. 0% B By factors 3 and 11. By factors 7 and 4. By factors 4 and 4. By factors 7 and 7. Don’t Know B 1. 2. 3. 4. 5.