Find a fundamental matrix of each of the systems in Problems1 through 8, then apply Eq. (8) to find a solution satisfying thegiven initial conditions.-3-2х,3|5. x'x(0) = |9.328. x' =-5-4-2X,x(0)53THEOREM 1Fundamental Matrix SolutionsLet (1) be a fundamental matrix for the homogeneous linear system x' = Ax.Then the [unique] solution of the initial value problemx' = Ax,x(0) = Xo(7)is given byx(1) = (1)(0)-Xo.(8)I found this answer in the book2 cos 3t-2 sin 3t5. $(t) =-3 cos 3t +3 sin 3t3 cos 3t +3 sin 3t3 cos 31 – sin 3tx(t) =%3D-3 cos 3t + 6 sin 3t[e3-e3el8. Ф(()e-21-ex(t)-e' +e-2%3D2re-2

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve the equation Ax = b by using the LU factorization given for A. Also solve Ax = b by ordinary row reduction. A = y = 4 - 7 - 4 3 12 - 5 X = 4 3 -9 = 1 -1 00 10 3 - 4 1 Let Ly = b and Ux=y. Solve for x and y. 4 -7 -4 0 -4 - 1 0 0 - 1 Row reduce the augmented matrix [A b] and use it to find x. The reduced echelon form of [A b] is yielding x = 4 - 15 61
2 17. Solve the matrix equation: -2X + 3 -8 4 -6 -4 2 2 -8 A. -5 -9 1 С. -9 -7 1 5 1 1 В. -9 1 15 -7 7 5 D.
Please answer number 3. Find the inverse of the following matrices using Adjoint Matrix Method
Q29. If A = , then the matrix is ((a)positive definite (b) negative definite (c) positive semi definite (d) negative semi definite (e) none * Consider to answer the following two questions. 1 a Q30. The system has unique solution if (a)a *j (b) a # -j (c) a #1 (d) a ± -1 (e) none Q31. The system has many solutions if (a) a =-j and B=-1 (b) a = -j and ß = 1 (d) a =-j and B =-j (e) none ) a + -1 (e) none (c) a = j and ß = -1 1+j] and y=|2-3j , then the value of a such that x Iy is 2j Q32. Suppose x= i (a) j (b) -j (c)) +j Q33.The smallest number of zeros in any 4 by 4 singular matrix is (a)4 (b) 6 (c) 8 (d) 10 (e) 2 Q34. Suppose the matrix A BC, where BeR and CeR rank(A) S4 (b) rank(A) S3 Q35.If the matrix A is n by n and 24=72 and (34)=3, then the value of n is (a)2 (b) 3 (c)4 (d)s (e) 6 Q36.If u and v are unit real vectors and u-2v 3, then the angle between them is ( in degrees) (a)30 (b) 45 (c)60 (d) 90 (e) 120 Q37.One of the following is a vector space (a)all n by n nonsingular…
(4) Given matrix A = ( 2 −1 3 5 ) , and that h(T) = T 2 − 2T − 7, find h(T). ??????? ??? ? (1) A course repeater claims that his brother who did some Calculus at college years ago, but currently on some cough medication, believes that the first derivative of f(x) = (2x 2 − 1)(x 2 + 3) x 2 + 1 is f ′ (x) = [ 4x 2x 2 − 1 + 2x x 2 + 3 − 2x x 2 + 1 ][ (2x 2 − 1)(x 2 + 3) x 2 + 1 ] and that the first derivative of g(x) = (x 2 + 8) 7 (2 − 3x2) 5 is g ′ (x) = [ 14x x 2 + 8 + 30x 2 − 3x2 ] [ (x 2 + 8) 7 (2 − 3x2) 5 ] Using the product, qoutient and chain rules you learnt in Unit 7 of this Course, could you confirm whether his brother is right, wrong or a little tipsy. (2) In the following triangle below, the degree measures of the three interior angles and two of the exterior angles are represented with variables and expressions. (a) Find the size of each of the interior angle.
(a) Use Gauss-Jordan elimination to find the inverse of the given matrix A 1 1 3 3 -1 1 A = -1 -3 -4 2 -5 2 (b) Given b = (1.1.2.2) solve the system by suing x = A-'b.
3. 1 Given the matrix D = 2 3 1 -3 -2 4 -2 " find the inverse matrix of D.
7. Let A be a matrix with a row consisting of all zeroes. If B is a matrix such that AB is defined, which of the following is always true? 8. Given the matrix B = 1 1 11 Which of the following (233) (a) 3 2 3 332 matrices solves X in the matrix equation X+13= 2(X - B)? (211) (d) 1 2 1 112 (322) (b) 2 3 2 2 2 3 (122) (c) 2 1 2 2 2 1
4. Consider the matrix equation Ax = b where, 4- (19) A= = where a, b > 0. (a) When is A strictly diagonally dominant and what does this tell us about the Gauss- Seidel method for Ax : b. = (b) Find the iteration matrix for the Gauss-Seidel method applied to Ax = b. (c) Using your previous answers, for what values of a and b does the Gauss-Seidel method converge.
What values for x and y are needed in the linear combination below to make the equation correct? 1 3 + y X =| 1 y =| 2 The system of equations 2x + y = 1, 3x – 4y = 7 can be expressed as the vector equation - 1 xu + YV using the vectors u = and v = The same system of equations can be written in a matrix form A b using the matrix A = and the vector b =
2. The augmented matrix for the system of equations 211 + 2r2 + I3 = 5, 2i + 3r2 + a3 = 2, 3r1 + 4r2 + 5z3 = 1, is given by O (2 2 1 1 4 5 2x 2x2 3r2 3z, 4r2 5z3 1 2 2 1 3. 1 2 4 5 1 '5 2 2 1 2. 3 4 5 1 1 1 213 213

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Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. -3 -2 х, 3 -[-] 5. х x(0)