a={{.9925,.0125},{.0075,.9875}}

Eigensystem[a]

eigenvalues=Eigensystem[a][[1]]
eigenvectors=Eigensystem[a][[2]]
MatrixForm[eigenvectors]

initial={10,990}

coeff=LinearSolve[Transpose[eigenvectors],initial]

t[n_]:=coeff[[1]](eigenvalues[[1]]^n)eigenvectors[[1]]+coeff[[2]](eigenvalues[[2]]^n)eigenvectors[[2]];
Plot[{t[n][[1]],t[n][[2]]},{n,1,150},PlotStyle->{Red,Green},PlotLegends->{"Type A", "Type B"}]