# Say A = [[-157,-72],[360,165]] is P^-1DP then A^n equals to?

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Given `A=[[-157,-72],[360,165]]=P^-1DP`

Here D is the diagonal matrix whose diagonal entries are eigen values of the matrix A.

Now in this case

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

Here we see that the eigen values of A are 3 and 5

So, `D^n=[[3,0],[0,5]]^n=[[3^n,0],[0,5^n]]`

So, `A^n=P^-1[[3^n,0],[0,5^n]]P` .

`A=P^(-1)DP` ,then

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

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

since matrix multiplication is associative. so we have

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

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

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