Explanation Given information: Given Y' ( t ) = A Y ( t ) , Y ( 0 ) = Y 0 where A ' = ( 1 − 2 3 − 4 ) Y 0 = ( 1 2 ) Calculation: A = [ 1 − 2 3 − 4 ] , Y 0 = [ 1 2 ] First, let’s find eigenvalues and corresponding eigenvectors of the matrix A in order to factor it into the product X D X − 1 . det ( A − λ I ) = | 1 − λ − 2 3 − 4 − λ | = ( 1 − λ ) ⋅ ( − 4 − λ ) − ( − 2 ) ⋅ 3 = λ 2 + 4 λ − λ − 4 + 6 = λ 2 + 3 λ + 2 = λ 2 + λ + 2 λ + 2 = λ ( λ + 1 ) + 2 ( λ + 1 ) = ( λ + 1 ) ( λ + 2 ) Therefore, λ 1 = − 1 , λ 2 = − 2 , D = [ − 1 0 0 − 2 ] For λ 1 = − 1 , we have ( A − λ 1 I ) x 1 = [ 1 − ( − 1 ) − 2 3 − 4 ( − 1 ) ] [ x 1 ( 1 ) x 1 ( 2 ) ] = [ 0 0 ] This gives us system of equations 2 x 1 ( 1 ) − 2 x 1 ( 2 ) = 0 ⇒ x 1 ( 1 ) = x 1 ( 2 ) 3 x 1 ( 1 ) − 3 x 1 ( 2 ) = 0 If we take x 1 ( 1 ) = 1 , we get x 1 ( 2 ) = 1 Therefore, corresponding eigenvector is given by x 1 = ( 1 , 1 ) T For λ 2 = − 2 , we have ( A − λ 2 I ) x 2 = [ 1 − ( − 2 ) − 2 3 − 4 ( − 2 ) ] [ x 2 ( 1 ) x 2 ( 2 ) ] = [ 0 0 ] This gives us system of equations 3 x 2 ( 1 ) − 2 x 2 ( 2 ) = 0 ⇒ x 2 ( 2 ) = 3 2 x 2 ( 1 ) 3 x 2 ( 1 ) − 2 x 2 ( 2 ) = 0 If we take x 2 ( 1 ) = 1 , we get x 2 ( 2 ) = 3 2 Therefore, corresponding eigenvector is given by x 2 = ( 1 , 3 2 ) T Thus matrix X is given by X = [ x 1 x

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

9th Edition

ISBN: 9780321962218

Author: Steven J. Leon

Publisher: PEARSON

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Chapter 6, Problem 7CTB

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To compute: etA and use it to solve the initial value problem.

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Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Ch. 6.1 - Find the eigenvalues and the corresponding...Ch. 6.1 - Show that the eigenvalues of a triangular matrix...Ch. 6.1 - Let A be an nn matrix. Prove that A is singular if...Ch. 6.1 - Let A be a nonsingular matrix and let be an...Ch. 6.1 - Let A and B be nn matrices. Show that if none of...Ch. 6.1 - Let be an eigenvalue of A and let x be an...Ch. 6.1 - Let A bean nn matrix and let B=I2A+A2. Show that...Ch. 6.1 - An nn matrix A is said to be idempotent if A2=A....Ch. 6.1 - An nn matrix is said to be nilpotent if Ak=O for...Ch. 6.1 - Let A be an nn matrix and let B=AI for some scalar...

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