Find the general solution ofthe system whose augmentedmatrix is given in the following,Q. No. 1[1 3 4 7113761b) Detemine the value(s) of h such that the matrix is the augmented matrix of a consistent linearsystem,[1 -36-21c) Find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the given[4-1]matrix,112.

Solution of the system of linear equation

Answered: Find the general solution ofthe system…

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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A₁ Find the eigenvalues and eigenvectors of the coefficient matrix A. +8) = x1 x2 Consider the following system: = dt = and X₂ = = -3 5 -5 -3 1 V₂ Note: When Webwork checks whether eigenvectors and eigenvalues are correct, ALL eigenvalues and eigenvectors are evaluated at once. So individual eigenvectors and eigenvalues won't be marked correct unless all the eigenpairs are correct. Sketch a phase portrait. Solve the system given initial conditions ₁ (0)=1, x₂(0) = -2. I
Consider the below 3 equations, U₁ = C₁-C₂ + C3 U₁= C₁ + (0*C₂)+ (0*C3) = C1 U₁ = C1 + C₂ + C3 In Matrix form, 1 -1 1] 10 1 Solve to get the below final form, C₁ 5 C3 || C₁ U₁ 0₁²₂}=- u, 1||C3 U₂ 12 U₁ (u₁-2u, +Ug) ¹ (Ug-u₁)
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Find the LU factorization of A = and then use it to solve the system -4 -3 3 16 15 -11 A = x1 = x₂ = x3 = 8 -6 -8 || x1 x2 x3 = -4 -3 3 16 15 -11 8 -6 -8 -31 127 44 " such that the matrix L has ones along the diagonal,
8. Solve the linear system Ax-b by x₁ + x₂-3x₂ = -1 -x₁ + 2x₂ = 1 x₁ - x₂ + x₂ = 2 a) finding the LU-factorization of the coefficient matrix A b) Solving the lower triangular system Ly-b. c) solving the upper triangular system Ux-y.
4. Consider the following matrix: 3. -2 A =-14 15 18 -18 -10 (a) Find the eigenvalues of A. (b) Find three linearly independent eigenvectors for A and state the algebraic and geometric multiplicity for each eigenvalue. Form a matrix C whose columns are the three linearly independent eigenvectors that you found in (b). Verify that C-1 AC is a diagonal matrix (which we call D). (d) Use your previous answer to find a matrix B with the property that B3 = A. (e) For any real matrix A, write down a sufficient condition for a real matrix E to exist such that E2 = A, and prove that this condition is sufficient. %3D

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