Find sequences x(t), y(t), z(t) satisfying the following system of linear difference equations:

x(t)=6x(t-1)+13y(t-1)-8z(t-1)
y(t)=2x(t-1)+5y(t-1)-2z(t-1) &initial conditions x(0)=1, y(0)=1, z(0)=0
z(t)=7x(t-1)+17y(t-1)-9z(t-1)
I have found the following
1 1/2 -1
0 1/2 1/2 is P
1 1 1

-2 0 0
0 3 0 is the diagonal matrix D
0 0 1

-1 -3 2
2 4 -2 is the inverse of P
1 1 1