To find the steady state for this 3 by 3 matrix, we need to find the values of x, y, and z such that the matrix equation equals the vector [x, y, z].
Setting up the equation:
0.8x + 0.14y + 0z = x
0.1x + 0.71y + 0.75z = y
0.1x + 0.14y + 0.25z = z
We can rearrange these equations to isolate x, y, and z:
0.8x - x = -0.14y
0.1x - y = -0.71y + 0.75z
0.1x - z = -0.14y + 0.75z
Simplifying further:
-0.2x = -0.14y
0.1x + 0.29y = 0.75z
0.1x + 0.14y - z = 0
Now, we can solve this system of equations. Since there are three unknowns and three equations, the solution will give us the values for the steady state.
Multiplying the first equation by 5 and the third equation by 10 to eliminate decimals:
-1x = -0.7y
x = 0.7y
Plugging this back into the second equation:
0.1(0.7y) + 0.29y = 0.75z
0.07y + 0.29y = 0.75z
0.36y = 0.75z
y = (0.75/0.36)z
Finally, plugging this back into the first equation:
x = 0.7y
x = 0.7(0.75/0.36)z
x = 1.458z
So, the steady state in terms of x, y, and z is:
x = 1.458z
y = (0.75/0.36)z
z = z
This means that in the steady state, the values of x, y, and z are all determined by the value of z.
in 3 by 3 matirx
0.8x+0.14y+0Z
0.1x+0.71y+0.75z
0.1x+0.14y+o.25z
find the stady state
1 answer