# 14add_2 - r a 1 2 du r r 1. Consider the equation = 0 a 3 u...

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1.Consider the equation1aWhat are the eigenvalues ofA? How many linear independent eigenvectors can youfind?1bCheck that(29[ ]30zero matrixAaI-==1c.Show that(29(222AtatAaIteeIAaI t-=+-+.1d.Solve (1) with the given initial condition (express your answer in terms of the constanta120300aduauAudta==rrrsubjected to the initial condition[](0)0,0,1Touu==rr..29).1e.What is the solution of (1) if?1f.What is the solution of (1) if(29(21cNAaI-r.Write(AI tAteeeλλ-=.Find1crand2crusing the initial condition[](0)0,0,1Touu==rr.It1b.Direct calculation shows that
(29(29[ ]2230120030120030030000030000000000000000AaIAaI  -==-==   1c.Using the fact thatIandAcommutes, we have(t A aIAtatIatIAtatIatIAtatIeeeeeeee---===NoteatIateeI=(as shown in class)(1a)(29(29(29(29[ ]23230[0]2!3!t A aItteIt AaIAaIAaI-=+-+-+-+14243(1b)In (1b), we used (1b), i.e.,(29[ ](29[ ]300nAaIAaI-=-=for3n.Combining(1a,b), we have(29(29(29(2922222!2AtatattteeIItAaIAaIeItAaIAaI=+-+-=+-+-1d.The solution of the initial value problem is(29(29220000200020120030030002000000AtatatteueutAaI uAaIuteutuu=+-+-=++rrrrrrrSubstituting[](0)0,0,1Touu==rrgives:20230302100attue

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