In Problems 5–9, find a fundamental matrix for the systemx' (t) = Ax(t) for the given matrix A.5-15. А %3D8.-2-26. А %3D-1-1-4||

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Answered: In Problems 5–9, find a fundamental…

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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13. Find all value of h such that the augmented matrix (h+1) (h+2) 0 0 (h-2) (h-2) 0 0 (h² - 1) represents a system with: (a) no solutions; (b) a unique solution; (c) infinitely many solutions.
If A is a constant matrix, the fundamental matrix of the system x'(t) = Ax(t) is non-constant. O True False
1. (a) Write the following system in matrix notation: x = 7x1 + 5x2 – 9x3 x2 = 4x1 + x2+ x3 %3D x'3 = - 2x2 + 3x3 (b) Write the following system without the use of matrices: X1 1 t+1 X1 + 4t2 1 x2 (c) Which, if any, of the systems is nonhomogeneous?
Problem 5 () (a) Given matrix S that O(A)S == x' = 2x - 3y y' = x + y ¹ ( ₁ ) = $(*) ² S cont. O(B)S=; 1 2 O(C)S = ( -1 2 O(D)S = (12) O(E)S=231 -1 basis? O(A) (1) 1 O(B) cont. 1 3 -1 2 1 1 (b) What are the basis vectors i' and ' in the original -1 1 and (²) ¹ (²3) and O(C) -1 and OD) (1) a OCE) (1) and (2) and (c) If a point P' = - (₂) co-ordinates in the original basis? . What is the transformation O(A)S== O(B)S = (3) O(C)S= O(D)S== (3) O(E)S = (¹) in the new basis, what is its cont. (2²) evaluate the matrix in the new basis. (d) A matrix M = O(A)M' = O(B)M' = O(C)M' = O(D)M' = O(E)M' = 5 0 0 10 10 0 0 6 (7 0 0 10/ 5 10 0 0 10 0 07 in the original basis,
* 5. Let 7' is a matrix of iterative method for solving linear system Ax-b. If A is a positive definite and tridiagonal matrix, then p(T) is equal to (b) [p(TG)] ² (d) None of the above (a) √p(TG) (c) p(TG) O a O b O C O d
D, E and F
(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.
9. P = 15 -4 -7 2e31 – 8e- -4e31 + 2e- ž(1) = | 3e3t – 20e- -6e31 + 5et Show that x1 (t) is a solution to the system x = Px by evaluating derivatives and the matrix product -4 ž(1) = | 15 -7 Enter your answers in terms of the variable t. Show that x2(t) is a solution to the system x' = Px by evaluating derivatives and the matrix product 9. 3(1) = | 15 -4 X2(t) -7 Enter your answers in terms of the variable t.
Problem 4. Let A = 3 1 (³₂2) 52 and B = (32). Compute A-¹ and use it to: (i) find a 2 × 2 matrix X such that AX = B, (ii) find a 2 × 2 matrix Y such that YA = B.
Example 4.5. If A and C are non-singular matrices, then sho A 0 B C c] = [-~^BA-² 0 &] C-1 1 0 0 0 0 Hence find inverse of 2 0 0 0 (E 3 04 0 1 0 3
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
(8) If A is strictly dominant matrix, then || A ll2= p(A).

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In Problems 5–9, find a fundamental matrix for the system x' (t) = Ax(t) for the given matrix A. 5 -1 5. A = 8. 2 -2 %D -1 -1 -4 |