Qn3 A =with the given eigenvalue A = i. Answer the following questions:a) Find the general solution to the system (AI – A)x = 0 for the provided value ofb) Find a basis vector x for the general solution obtained in Question (a)c) Hence, find an invertible matrix P and a matrix C such that A = PCP-1.

Name: Qn3 A =with the given eigenvalue A = i. Answer the following questions
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Description: Qn3 A =with the given eigenvalue A = i. Answer the following questions:a) Find the general solution to the system (AI – A)x = 0 for the provided value ofb) Find a basis vector x for the general solution obtained in Question (a)c) Hence, find an inver

(a) Given : A=1-21-1, λ=i consider λI-Ax=0 ⇒iI-Ax=0⇒i-12-1i+1x1x2=0⇒-x1+i+1x2=0⇒x1=i+1x2 So, general…

Answered: A = with the given eigenvalue A = i.…

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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Qn3

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Step 1

(a) Given : $A = [\begin{matrix} 1 & - 2 \\ 1 & - 1 \end{matrix}], λ = i$

consider $(λ I - A) x = 0$

$\Rightarrow (i I - A) x = 0 \Rightarrow [\begin{matrix} i - 1 & 2 \\ - 1 & i + 1 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = 0 \Rightarrow - x_{1} + (i + 1) x_{2} = 0 \Rightarrow x_{1} = (i + 1) x_{2}$

So, general solution of $x$ is

$x = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} i + 1 \\ 1 \end{matrix}] x_{2}$

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