APL (22)
{⍵+.×⍨2 2⍴3 5 1}⍣⎕⍨2↑1
Explanation:
{...}⍣⎕⍨2↑1: read a number, and run the following function that many times, using[1,0]as the initial input.2 2⍴3 5 1: the matrix[[3,5],[1,3]]⍵+.×⍨: multiply the first number in ⍵ by 3, the second by 5, and sum them, this is the new first number; then multiply the first number in ⍵ by 1, the second by 3, and sum those, that is the new second number.