Find the steady-state vector for the matrix below.

$P = \begin{bmatrix} 0.3 & 0.2 & 0.2 \\ 0.5 & 0.6 & 0.5 \\ 0.2 & 0.2 & 0.3 \end{bmatrix}$

(Type an integer or simplified fraction for each matrix element.)

Neetesh Kumar | October 18, 2024

Linear Algebra Homework Help

This is the solution to Math2B Course: Linear Algebra
Assignment: Ch5 Section 9 Question Number 4
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Step-by-step solution:

To find the steady-state vector, we need to solve for $q = \begin{bmatrix} q_1 \\ q_2 \\ q_3 \end{bmatrix}$, which satisfies the equation:

$P \mathbf{q} = \mathbf{q}$

For the given matrix:

$P = \begin{bmatrix} 0.3 & 0.2 & 0.2 \\ 0.5 & 0.6 & 0.5 \\ 0.2 & 0.2 & 0.3 \end{bmatrix}$

This leads to the system of equations:

$0.3 q_1 + 0.2 q_2 + 0.2 q_3 = q_1$
$0.5 q_1 + 0.6 q_2 + 0.5 q_3 = q_2$
$0.2 q_1 + 0.2 q_2 + 0.3 q_3 = q_3$

Step 1: Rearrange the system of equations

Rearranging each equation:

For the first equation:

$0.3 q_1 + 0.2 q_2 + 0.2 q_3 = q_1$

$0.3 q_1 + 0.2 q_2 + 0.2 q_3 - q_1 = 0$

$-0.7 q_1 + 0.2 q_2 + 0.2 q_3 = 0$
(Equation 1)

For the second equation:

$0.5 q_1 + 0.6 q_2 + 0.5 q_3 = q_2$

$0.5 q_1 + 0.6 q_2 + 0.5 q_3 - q_2 = 0$

$0.5 q_1 - 0.4 q_2 + 0.5 q_3 = 0$
(Equation 2)

For the third equation:

$0.2 q_1 + 0.2 q_2 + 0.3 q_3 = q_3$

$0.2 q_1 + 0.2 q_2 + 0.3 q_3 - q_3 = 0$

$0.2 q_1 + 0.2 q_2 - 0.7 q_3 = 0$
(Equation 3)

Step 2: Use the condition that $q_1 + q_2 + q_3 = 1$

Since the steady-state vector must sum to 1:

$q_1 + q_2 + q_3 = 1$
(Equation 4)

Step 3: Solve the system of equations

We now have the following system of equations:

$-0.7 q_1 + 0.2 q_2 + 0.2 q_3 = 0$
$0.5 q_1 - 0.4 q_2 + 0.5 q_3 = 0$
$0.2 q_1 + 0.2 q_2 - 0.7 q_3 = 0$
$q_1 + q_2 + q_3 = 1$

Solving this system of equations gives the steady-state vector $q$.

Final Answer

$q = \begin{bmatrix} \frac{2}{9} \\ \frac{5}{9} \\ \frac{2}{9} \end{bmatrix}$

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