Z Transform How to Handle the Inital Condition
Using the Z-transform table on this website

Z-Transform Table Section 9.2



The forward tap, item # 7, has the initial condition, which means for a zero input response, x[k] can be calculated to account for the initial condition in the z domain. 

In this case, I was trying to calculate: Z{x[k+1] = (I+TA)x[k]}.

yields-> z(X(Z) - x(0)) = (I+TA)X(Z)

Solving for X(Z) = z*x(0) / [z-(I+TA)]

Solving x[k] = Z^-1{X(Z)} = x(0) * (1+TA)^k