Explanation The given matrix is A = ( 2 1 1 2 ) . Find the eigenvalue of A . det ( A − λ Ι ) = 2 − λ 1 1 2 − λ = λ 2 − 4 λ + 3 = ( λ − 3 ) ( λ − 1 ) Equate above equation to 0 gives, ( λ − 3 ) ( λ − 1 ) = 0 λ = 1 , 3 , Thus, the eigenvalues are λ 1 = 1 and λ 2 = 3 . The eigenvector corresponding to λ 1 = 1 is computed as follows. ( 2 − 1 1 1 2 − 1 ) ( x y ) = ( 0 0 ) ( 1 1 1 1 ) ( x y ) = ( 0 0 ) ( x + y x + y ) = ( 0 0 ) Which is equivalent to the system of equations, x + y = 0 x + y = 0 Solve the above equations and obtain y = − x . Therefore, the eigenvector corresponding to λ 1 = 1 is K 1 = ( − 1 1 ) . The eigenvector corresponding to λ 2 = 3 is computed as follows. ( 2 − 3 1 1 2 − 3 ) ( x y ) = ( 0 0 ) ( − 1 1 1 − 1 ) ( x y ) = ( 0 0 ) ( − x + y x − y ) = ( 0 0 ) Which is equivalent to the system of equations − x + y = 0 x − y = 0 Solve the above equations and obtain x = y . Therefore, the eigenvector corresponding to λ 2 = 3 is K 2 = ( 1 1 ) . Hence, the Corresponding eigenvectors are, Let P = ( − 1 1 1 1 ) denote a matrix whose columns are eigenvectors ( K 1 and K 2 ) corresponding to the eigenvalues ( λ 1 and λ 2 )

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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Chapter 8.4, Problem 20E

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Chapter 8.4, Problem 19E

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Solution Summary: The author explains the eigenvalues of the given matrix: A = (cc2& 1 1& 2end

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