In this problem a "state vector" is also a probability vector, that is, it is a column vector with non-negative entries whose sum is 1.

Let $A = \left[\begin{array}{cccc} 0.5 &0 &0 &0.5\cr 0 &1 &0 &0\cr 0 &0 &1 &0.5\cr 0.5 &0 &0 &0 \end{array}\right]$.

Use computational experimentation to find two initial state vectors (they must also be probability vectors) which lead to different steady-state vectors for the matrix $A$. (NOTE: Even if it is less computationally efficient, it may be easier to compute steady state vectors using powers of A instead of the recursive formula.)

$x_0 =$

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$y_0=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$