Program Explanation Given information: The coefficient matrix has been given x ' = [ 3 − 4 1 0 4 3 0 1 0 0 3 − 4 0 0 4 3 ] x Explanation : Consider the following initial value problem x ' = [ 3 − 4 1 0 4 3 0 1 0 0 3 − 4 0 0 4 3 ] x The given system has been written into the matrix form x ' = A x Where coefficient matrix (A) is A = [ 3 − 4 1 0 4 3 0 1 0 0 3 − 4 0 0 4 3 ] Characteristic equation of the given coefficient matrix is x 4 - 12x 3 + 86x 2 - 300x + 625 = 0 And the eigenvalues are λ = 3+ 4i, 3 - 4i, 3 + 4i, 3 - 4i The eigenvalues are complex conjugate with positive real part Using MATLAB code to find the general solution of the coefficient matrix x 1 ( t ) = 42 × C 33 × exp(t) - (76 × C 34 × exp(t))/49 + C 36 (exp(t) + 21 × t 2 × exp(t) + 34 × t × exp(t)) + … C 35 (34 × exp(t) + 42 × t × exp(t)) x 2 ( t ) = 7 ×

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Question

Chapter 5.5, Problem 33P

Program Plan Intro

Write a code to find the characteristic equation of the given coefficient matrix.

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Chapter 5.5, Problem 32P

Chapter 5.5, Problem 34P

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| IF A = 3 -4 3 0-3 -2 The Cofactor Matrix is :-

Determined the eigenvector of the matrix: please print A=[1 0 4, 0 4 0, 3 5 -3]

(Cayley-IHamilton Theorem) The Cayley-Ilamilton theorem is a powerful theorem in Linear algebra that states: every square matrix of real numbers satisfies its own char- acteristic equation. For 2 x 2 matrices this can be seen in the following way. First, we introduce the trace of the matrix tr A 41,1 +22 and the determinant det A = a11a22 - 41,22,1- Then, the chacteristic polynomial is the quadratic given by p(A) = x – (tr A)A + (det. A). Then, for any 2 x 2 matrix, if you plug it into this polynomial, you should always get a zero matrix. c_h_test Function: Input parameters: • a single 2 x2 matrix for which you would like to test the Cayley Ilamilton Theorem Local Variables: • two scalars to represent the trace and determinant Output parameters: a single 2 x2 which is the result of evaluating the charcteristic polynomial at A (note: think carefully about the constant term) A possible sample case is: > mat_B = c_h_test([3, 2 ; 2, 0]) mat B = 0 0

Chapter 5 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2P Ch. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4P Ch. 5.1 - Prob. 5P Ch. 5.1 - Prob. 6P Ch. 5.1 - Prob. 7P Ch. 5.1 - Prob. 8P Ch. 5.1 - Prob. 9P Ch. 5.1 - Prob. 10P

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Write a code to find the characteristic equation of the given coefficient matrix.

Solution Summary: The author explains how to write a code to find the characteristic equation of the given coefficient matrix.

Write a code to find the characteristic equation of the given coefficient matrix.

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Chapter 5 Solutions