I don't understand how the following equation is done:

I understand how to calculate the characteristic polynomial, which isGiven is a matrix that represents how coins are circulating each day:

P: 60% stay at P, 10% go to Q, 30% go to R

Q: 10% go to P, 80% stay at Q, 10% go to R

R: 10% go to P, 20% go to Q, 70% stay at R

C: ((0.6,0.1,0.3),(0.1,0.8,0.1),(0.1,0.2,0.7))

Calculate the equilibrium division, as the number of days reaches infinity

I understand how to calculate the eigenvalues and -vectors, which are-λ^3+2.1λ^2-1.4λ+0.3

I understand how to calculate the transition matrix and the inverse (which is simply the eigenvectors)λ = 1 : (4,9,7)

λ = 0.6 : (0,-1,1)

λ = 0.6 : (-1,-1,2)

But now comes the part I do not understand:T = (((1/20),(25/20),(-16/20)),((1/20),(-15/20),(4/20)),((1/20),(5/20),(4/20)))

T^-1 = ((4,9,7),(0,-1,1),(-1,-1,2))

I get how the matrix is formed (because when you near infinity, all values below 1 get reduced to 0). But when you multiply by this matrix, you will get a matrix like:lim C^n = T^-1 * ((1,0,0),(0,0,0),(0,0,0)) * T

n ->∞

((a,b,c),(0,0,0),(0,0,0)).

In stead, the answer in our book is:

I dont understand how they can multiply by that matrix and still get that answer.(((4/20),(9/20),(7/20)),((4/20),(9/20),(7/20)),((4/20),(9/20),(7/20)))

Any help would be much appreciated.