Final answer:
To create a stochastic matrix that has the given steady-state vector, one must choose values that cause the columns to sum to 1 and satisfy the steady-state condition when multiplied by the vector.
Step-by-step explanation:
To find a 2 x 2 stochastic matrix A that has a steady-state vector [9/13 4/13], we must remember that a stochastic matrix has the property that each column sums to 1, and when multiplied by a steady-state vector, it yields the vector itself. Therefore, let's assume our matrix A has the form:
A = |a b|
|c d|
Stochastic conditions require that the following must be true:
a + c = 1
b + d = 1
For the steady-state condition, we should have:
A * [9/13] = [9/13]
[4/13] [4/13]
This results in the system of equations:
a*(9/13) + b*(4/13) = 9/13
c*(9/13) + d*(4/13) = 4/13
From these conditions, you can choose values for a, b, c, and d that satisfy both the stochastic and steady-state conditions. An example could be:
A = |9/13 2/13|
|4/13 11/13|
Here, both columns sum up to 1, and when this matrix is multiplied by the steady-state vector, it yields the vector itself.