Page rank via eigen vectors :

A={{0,0,1/2,1},
{1/3,0,0,0},
{1/3,1/2,0,0},
{1/3,1/2,1/2,0}};
Eigenvalues[A]

Eigensystem[A]

x={12,4,6,9}/31 //N

Page rank via dynamical system

v={1/4,1/4,1/4,1/4};
A.v//N

A.A.v//N

MatrixPower[A,4].v//N

MatrixPower[A,8].v//N