Markov's Chain stationary matrix determination

I have the following transition matrix $P$

begin{equation}
P=
begin{pmatrix}
1/2 & 1/3 & 1/6\
1/4 & 3/4 & 0\
1/5 & 2/5 & 2/5
end{pmatrix}
end{equation}

I have tried to calculate the stationary matrix as
begin{equation}
P^n= C D^n C^{-1}
end{equation}

With $D=C^{-1} P C$ the diagonal matrix. Calculating $C$ with the eigenvalues and eigenvectors

begin{equation}
C=
begin{pmatrix}
v_1 & w_1 & z_1\
v_2 & w_2 & z_2\
v_3 & w_3 & z_3
end{pmatrix}
end{equation}

where the eigenvectors are

begin{align}
vec{v} =langle (v_1,v_2,v_3)rangle &= langle(1,1,1)rangle \
vec{w} =langle(w_1,w_2,w_3)rangle &= langleleft(frac{2isqrt{39}+3}{6},frac{-3isqrt{39}-7}{16},1right)rangle \
vec{v} =langle(v_1,v_2,v_3)rangle &= langleleft(frac{-2isqrt{39}+3}{6},frac{3isqrt{39}-7}{16},1right)rangle \
end{align}

so the diagonal matrix is
begin{equation}
D=
begin{pmatrix}
1 & 0 & 0\
0 & frac{-isqrt{39}+39}{120} & 0\
0 & 0 & frac{isqrt{39}+39}{120}
end{pmatrix}
end{equation}

how can i calculate $D^{n}$? is the diagonalization correct?

I tried by other way using $BP=B$ where B is the final vector state, then

begin{align}
[p_1 quad p_2 quad p_3] begin{pmatrix}
1/2 & 1/3 & 1/6\
1/4 & 3/4 & 0\
1/5 & 2/5 & 2/5
end{pmatrix} = [p_1 quad p_2 quad p_3]
end{align}

using Gauss-Jordan elimination for solving this system it gives me infinite solutions.

I dont know if it is a problem with the transition matrix.