There are four adjacent squares and a marker. A player begins a game by placing a marker in square#2. A die is rolled and the marker is moved one square to the left if a 1 or 2 is rolled and one square to the right if a 3, 4, 5, or 6 is rolled. The process is continued until the marker lands in square#1 (win) or square#4 (lose). What is the probability of winning?
Hint: instead of diagonalizing the appropriate transition matrix, A, it is easier to represent e2 as a linear combination of the eigenvectors of A and then apply to the result.
I know how to answer this the "harder" way by diagonalizing A but I don't understand what the hint is saying. I assume e2 is the second vector in the standard ordered basis for 4x1 matrices but how do I "apply to the result" and why would I want to??