matrix algebra question?

Question

given the 2 x 2 matrix - A

( 0.9 0.05 )
( 0.1 0.95 )

please help me find A ^ (n ) where n is a positive integer

the intial condition is:
( v ) = 300
( r ) 100

please help me, I'm really lost

Answer 1

Diagonalize the matrix.

If your original matrix is A, then D=P^(-1) A P will be diagonal, where P is the matrix whose columns are eigenvectors.

Then D^n = P^(-1) A^n P. solve this for A^n by multiplying on the left by P and multiplying on the right by P^(-1).

I have no idea what v and r stand for, but if this is a discrete dynamical system, mutliply the matrix you get above by the column vector containing your initial values, and you will get the values of v and r after n iterations.