20.Consider a 2 × 2 system x′ = Ax. If we assume that r1 ≠ r2, the general solution is x=c1ξ(1)er1t+c2ξ(2)er2tx=c1ξ1er1t+c2ξ2er2t, provided that ξ(1) and ξ(2) are linearly independent. In this problem we establish the linear independence of ξ(1) and ξ(2) by assuming that they are linearly dependent and then showing that this leads to a contradiction.a.Explain how we know that ξ(1) satisfies the matrix equation (A − r1I)ξ(1) = 0; similarly, explain why (A − r2I)ξ(2) = 0.

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Answered: 20.Consider a 2 × 2 system x′ = Ax. If…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 31E

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20.Consider a 2 × 2 system x′ = Ax. If we assume that r₁ ≠ r₂, the general solution is x=c1ξ(1)er1t+c2ξ(2)er2tx=c1ξ1er1t+c2ξ2er2t, provided that ξ⁽¹⁾ and ξ⁽²⁾ are linearly independent. In this problem we establish the linear independence of ξ⁽¹⁾ and ξ⁽²⁾ by assuming that they are linearly dependent and then showing that this leads to a contradiction.

a.Explain how we know that ξ⁽¹⁾ satisfies the matrix equation (A − r₁I)ξ⁽¹⁾ = 0; similarly, explain why (A − r₂I)ξ⁽²⁾ = 0.

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