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

Expert Answers
rakesh05 eNotes educator| Certified Educator

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


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` .

pramodpandey | Student

`A=P^(-1)DP`   ,then


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

since matrix multiplication is associative. so we have