What are the limits as k -> infinity of the following:
[.4 .2; .6 .8]^k[1; 0]
[.4 .2; .6 .8]^k[0; 1]
[.4 .2; .6 .8]^k
I think the first two go to [1; 0] and the last goes to [0 0; 0 0], but am not sure.
The more complete method of solution involves diagonalizing the matrix. Let's say that for a matrix $\displaystyle A$ you could find an invertible matrix $\displaystyle P$ and a diagonal matrix $\displaystyle D$ such that $\displaystyle A=PDP^{-1}.$ Then
$\displaystyle A^{2}=(PDP^{-1})(PDP^{-1})=PDDP^{-1}=PD^{2}P^{-1},$
$\displaystyle A^{3}=(PDP^{-1})(PDP^{-1})(PDP^{-1})=PDDDP^{-1}=PD^{3}P^{-1}.$
More generally,
$\displaystyle A^{k}=PD^{k}P^{-1}.$
But the kth power of a diagonal matrix is just the (diagonal) matrix with the diagonal elements of the original diagonal matrix raised to the kth power.
So you should be able to compute the limit exactly.