Problem #5: Let A and B be n x n matrices. Which of the following statements are always true? (i) If det(A) = det(B) then det(A B) =0. (ii) If A and B are symmetric, then the matrix AB is also symmetric. (iii) If A and B are skew-symmetric, then the matrix A7 + B is also skew-symmetric. (A) none of them (B) (i) and (ii) only (C) (i) and (iii) only (D) (iii) only (E) all of them (F) (ii) only (G) (i) only (H) (ii) and (iii) only Problem #5: | Just Save ' ' Submit Problem #5 for Grading Problem #5 | Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #6: Find the eigenvalues of the following matrix. 4 -6 4 A=|0 -5 0 2 -6 -2 Problem #6: I:I separate your answers with a comma | Just Save ' ' Submit Problem #6 for Grading Problem #6 | Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #7: Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x. -7 -8 -8 2 A= | * * * | x=|-6 * * *# -8 Given that x is an eigenvector of the matrix A, what is the corresponding eigenvalue? | Just Save ' ' Submit Problem #7 for Grading Problem #7 | Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #8: Consider the following matrix. Given that ^ = 0 is an eigenvalue of A, find a basis for the eigenspace corresponding to ) = 0. Problem #8: Select V Just Save Submit Problem #8 for Grading Problem #8 Attempt #1 Attempt #2 Attempt #3 Your Answer Your Mark: Problem #9: Consider the following matrix. Find a matrix P that diagonalizes A. Problem #9: Select V Just Save Submit Problem #9 for Grading Problem #9 Attempt # 1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #10: Suppose that a 2 x 2 matrix A has eigenvalues A = -2 and 3, with corresponding eigenvectors [to ] and _'s , respectively, Find A2. Enter your matrix by row, with entries separated by Problem #10: Enter your answer symbolically, commas. as in these examples e.g., a b would be entered as a,b,c,dDATE Problem - 5 - you are giving several statements about metrices and their properties. . " if det (A) = det (B ) then det ( p - B ) = 0. false conte reexample: 0 Let A = and 1 = 1 Both A And & have determinants of I,but A - B = . and dot ( A - B ) = def I - 1 = 0 . Thus, this statment is false. statment (ii) - . " if A and is are symmetric, then the matrix AB is also symmetric . "false. counterexample .. Let A - 2 3 2 S S Both B AB = 9 IL symultiis bat which Theis . this statment is false . statment ( 111 ) - . " if A and B arce skew - symmetric , twin the metrix A + B is abo . Skew - symmetric . " True . for any skew - symmetric matrix A, we have AT = - A 80 AT + B = - AtB = - ( A-B ) . since the sum or difference of Skew-symmetric matrices re ATtD is also skewe - symmetric skew - symmetric . They this statment in thece. correct Anwere . D (isi ) onlyPAGE NO. 3 DATE Problem - 6 Px To find the eigenvalues of matrix A= we need to solve the characteristic equation. det ( A - ) I ) = 0 Let 0 - X 5 - 6 - 2 = 10 - x ) ( - 6 - 2 ) - ( - 25 1-5) 1+ 6 1 + 10 = 0 solving the quadratic equation 1+ 6x t 10 = 0 ugly the quadratic formula, - 6+ /36 - 40 - 6 +1-4 2 2 2 ( 1) - 6+ 21 - 3+ 1 They . The eigenvalues of A arre -3+1 -3-1.Problem - 7 E19 6 Given n = an eigenvector of an unknown with finding matrix A, we are tayked the corresponding eigenvalue. if a is am eigenvector then . An = )K . since the second and third row of are not given . we use the fact that The matrix - vector Product in 4 scalar multiple of n leading to the eigenvalue! A . -- B 8 given this condition . The eigenvalue is be * = - 1 foundAoblong B - given matrix A = and orgenvalue 1=0 6 we are tasked with finding a basis for the eigenspace corresponding to 1 20 we need to find the nullspace of . A - OT = A : A = you reducing Ai some reduction : -7 O The general solution is JC = + 10 Theys, 9 basis for the eigenspace corresponding - 1 OProblem - 9 we are tasked with diagonalizing matrix A = from problem By 1 = 0 1 1 we already eigenvalue know that with multipli- 2. the other eigenvalue. can be found - city by wring ng the trace of A : + + ( A ] = 0 + 1 + ) - 2 They , the other eigenvalue is 1 = 2 . The elgeenvector corresponding to x = 2 is we can move construct the matrix p using the eigenrectors of columns ! - 1 L 7 P 1 - 2 0 The diagonal matrix o is They P diagonalizes A.Problem - 10 . given A is a axe metrix with ofjenaley AL - - 2 and 12 =3 and corresponding expenvectors 1 and via we are tasked with finding AZ we can diagonalize A as A = POP whine - 2 0 P = 10-5 , 0 3 The inverse of P is' I - 5 - 1 1/3 1 15 +5 - 10 L 2/2 - 415 mou, we can compute AZ = PD 2 p- 1 9 finally multiplying the metrixy , A = P. D 2. P - 40/3 - 1 3 Thees 4/3 1/15 - 1/3