Consider the matrix D = [[3,0],[0,5]] and P = [[9,4],[-20,-9]] Say A = [[-157,-72],[360,165]] is (P^-1)(D)(P) then A^n equals to?

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pramodpandey | College Teacher | (Level 3) Valedictorian

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`A=P^(-1)DP`    given

`A^n=(P^(-1)DP)^n`

`A^n=(P^(-1)DP)(P^(-1)DP)...........(P^(-1)DP) `    ,n times

`=P^(-1)D(PP^(-1))D(PP^(-1))D........D(PP^(-1))DP`

``   ( Matrix multiplication is associative)

`=P^(-1).D.D........DP`

`=P^(-1)D^nP`

`=P^(-1)([[3,0],[0,5]])^nP`

`=P^(-1)[[3^n,0],[0,5^n]]P`

But   `P=[[9,4],[-20,-9]]`

`|P|=-81+80=-1`

`P^(-1)=(-1)[[-9,-4],[20,9]]=P`

`A^n=[[81.3^n-80.5^n,36.3^n-36.5^n],[-180.3^n+180.5^n,-80.3^n+81.5^n]]`

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