Maybe this?
Eigenvalues[{{6/10, a}, {2/3, 0}}] Solve[%[[2]] == 1]
You can make an interactive simulation:
Manipulate[ DiscretePlot[ MatrixPower[{{6/10, a}, {2/3, 0}}, n] . {110, 30}, {n, 0, 10}, Joined -> True], {{a, 3/5}, 0, 1}]
