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

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.