The Ultimate Matrix

Let the results be

$\begin{bmatrix} v_1 \\ v_2\\v_3\\v_4 \end{bmatrix}$

where $v_1,v_2,v_3,v_4$ are row vectors.

Pre-multiply with the base matrix:

$\left\{ \begin{aligned} \frac{1}{2}v_1+\frac{1}{2}v_2=v_1 \\ \frac{1}{2}v_2+\frac{1}{2}v_3=v_2 \\\frac{1}{2}v_3+\frac{1}{2}v_4=v_3\\\frac{1}{4}v_1+\frac{1}{4}v_2+\frac{1}{2}v_4=v_4 \end{aligned}\right. \Rightarrow v_1=v_2=v_3=v_4=v$

Let $v = [x_1,x_2,x_3,x_4]$

Post-multiply with the base matrix: $\left\{ \begin{aligned} \frac{1}{2}x_1+\frac{1}{4}x_4=x_1 \\ \frac{1}{2}x_1+\frac{1}{2}x_2+\frac{1}{4}x_4=x_2 \\\frac{1}{2}x_2+\frac{1}{2}x_3=x_3\\\frac{1}{2}x_3+\frac{1}{2}x_4=x_4 \\x_1+x_2+x_3+x_4=1\end{aligned}\right. \Rightarrow \left\{\begin{aligned} x_1=\frac{1}{7} \\ x_2=x_3=x_4=\frac{2}{7} \end{aligned}\right.$

The last equation is due to the fact that the product of transition matrices is still a transition matrix.

I use the python code below, and yields the answer $\boxed7$

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class Matrix:
    def __init__(self, array_list):
        self.array_list = array_list
        self.row = len(array_list)
        self.column = len(array_list[0])
        for i in range(1, self.row):
            if len(array_list[i]) != self.column:
                raise IndexError
    def __mul__(self, other):
        if self.column != other.row:
            print ('Matrices can\'t be multiplied')
            print ('Error: First matrix\'s column number isn\'t equal to second matrix\'s row number.')
        list_answer = []
        for i in range(self.row):
            list_a_row = []
            for j in range(other.column):               
                element = 0
                for k in range(self.column):                    
                    element = element + self.array_list[i][k] * other.array_list[k][j]
                list_a_row.append(element)
            list_answer.append(list_a_row)
        return Matrix(list_answer)
    def visualize(self):
        row = self.row
        for i in range(row):
            print ('|' + str(tuple(self.array_list[i])) + '|')
    @classmethod
    def matrix_generator(cls, row, column):
        list_ = []
        for i in range(row):
            row_list = []
            for j in range(column):
                row_list.append(random.randint(-10,10))
            list_.append(row_list)
        return cls(list_)
mat = Matrix([[0.5,0.5,0,0],[0,0.5,0.5,0],[0,0,0.5,0.5],[0.25,0.25,0,0.5]])
def mat_exponent(mat, exp):
    cache = mat
    for i in range(2,exp+1):
        cache = cache*mat
    return cache
mat_exponent(mat,1000).visualize()
Output-
|(0.14285714285714285, 0.2857142857142857, 0.2857142857142857, 0.2857142857142857)|
|(0.14285714285714285, 0.2857142857142857, 0.2857142857142857, 0.2857142857142857)|
|(0.14285714285714285, 0.2857142857142857, 0.2857142857142857, 0.2857142857142857)|
|(0.14285714285714285, 0.2857142857142857, 0.2857142857142857, 0.2857142857142857)|