In Problems 21–24 verify that the
24.
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Chapter 8 Solutions
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- 3.16. Let P-1 and 2-11] = Q = Find a 2 x 2 matrix X such that PXQ: = -4 1] [2]arrow_forwardIn Problem. -24, the given vector functions are solutions to a system x' (t) = Ax(t). Determine whether they form a fundamental solution set. If they do, find a fundamental matrix for the system and give a general solution. -E]. et 24. x₁ = X2 sint COS [ -sint_ X cost sin/ COSTarrow_forward10. Find the general solution of the system of differential equations 3 -2 -2 d. X = -3 -2 -6 X dt 3 10 1 + 2tet + 3t?et + 4t°et 3 1 -3 Hint: The characteristic polymomial of the coefficient matrix is -(A- 4)²(A- 3). Moreover (:) 2 1 Xp(t) = t²et +t³et +t'e3t -1 -1 -3 is a particular solution of the system.arrow_forward
- 2. factor 1/4, then reflects about the line y = x. 01 1/4 0 0 1/4 (a) A = (d) A = (₁ Find the standard matrix for the operator on R² which contracts with ( :) :) 1/4 0 0 1/4 (1¹4) (e) None of these (b) A = 0 1 -(;!) 0 (c) A =arrow_forward6. Find the general solution of the system of differen- tial equations 6 -1 1 1 d 6t X - e 6. -1 1 2 dt 1 1 3 Hint: The characteristic polynomial of the coefficient matrix is –(A – 7)²(A – 4). | 2.arrow_forwardWhat can you conclude about the values of the quadratic form Q(x)?arrow_forward
- 7. Show that -{( ) ( !) (; :)} -1 2 6 9 S = 3 2 1 is linearly dependent in M23. Write one of the vectors as a linear combination of the others. 8. Find a basis for the solution space of Iị + x2 - X3 + 2x4 = 0 %3D X4 = 0 3x1 + 4x2 – 2x3 + 5x4 = 0 12 + 13 - -arrow_forward5. Find the cofactors and the adjoint matrix of A where (1 1 -1 b) A =| 2 3 (7 0 -1 a) A = - 2 0 - 2 4 -3 5 4 5 6 2arrow_forwardThis is the first part of a four-part problem. Let P = 2e3t – 6e -4e3t + 2e 1(t) = [3et 2(t) = -6e3t + 5e] 15et a. Show that j1(t) is a solution to the system i' = Pỹ by evaluating derivatives and the matrix product 9 = 15 Enter your answers in terms of the variable t. b. Show that ğa(t) is a solution to the system j' = Pj by evaluating derivatives and the matrix product = Enter your answers in terms of the variable t. 8 ]- [8 ]arrow_forward
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