liquid problem using linear algebra This was given to me as problem in a linear algebra class:
We are given two jars, the first containing 1 liter of liquid $A$ and the second 1 liter of liquid $B$. Also provided is a cup which has a capacity of $k$ liters, where $0 < k < 1$. We first fill the cup from the first jar and transfer the content to the second jar, stirring thoroughly afterwards. Next we dip the cup in the second jar and transfer $k$ liters of liquid back to the first jar. This operation is repeated again and again. Prove that as the number of iterations n of the operation tends to infinity, the concentrations of $A$ and $B$ in both jars tend to equal each other.

The expectation is to prove it using explicit concepts of linear algebra. I am totally lost...
Edit: I suppose it has to do with diagonalizable matrices?
 A: There are different way how to do that, even within the scope of linear algebra. Since the total amount of liquid is not changing and there are only two jars throughout the experiment, I recognize a Markov Process. There are two states, $A$ and $B$. Each time frame, there is $k$ chance that a molecule is transferred from one jar to the other and thus $1-k$ chance that "liquid" stays behind. The transition matrix is then $M=\pmatrix{1-k&k\\k&1-k}$. I do not know whether or not you have already dealt with eigenvalues, but typically in a closed Markov process, the dominant eigenvalue is $1$, and thus a corresponding eigenvector holds the information about the Steady State Vector. So her we go:$\pmatrix{1-k&k\\k&1-k}\pmatrix{x\\y}=1\pmatrix{x\\y}$. Working out this system results in two equations: $$(1-k)x+ky=x$$
$$kx+(1-k)y=y$$
When you work out these parentheses, you see that those equations are identical (as they should be, why?), so that's $y=x$ and so an eigenvector I easily found: $\pmatrix{1\\1}$. Any Initial State Distribution (the liquids in jar $A$ and $B$ as we start the experiment) could be represented by a vector $\pmatrix{a\\b}$. We can express this vector as a linear combination of the eigenvectors of $M$, say $\pmatrix{a\\b}=pv_1+qv_2$, with $p,q$ some scalars. (Have you covered the fact that eigenvectors typically are lin. independent and thus form a basis?) When we perform many times matrix $M$ onto this Initial State vector, then we get: $M^\infty$$\pmatrix{a\\b}$=$M^\infty$$(pv_1+qv_2)$. But $M^{\infty}v_1=\lambda_1^{\infty}v_1$ and $M^{\infty}v_2=\lambda_2^{\infty}v_1$. Since $\lambda_1$ is assumed the dominant eigenvalue, its eigenvector will ultimately "survive". Since the entries of this eigenvector are equal, as earlier established, the mixtures will be "equally" distributed. Note: I skipped around with some linear algebra material here and there, but if you are taking a course in this field, you should recognize it from class. Hope this helps...
