# Tutorial 3 Question 5 a)

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asked Oct 30, 2016
edited Oct 30, 2016

I have uploaded the question and my attempt. If someone could please let me know if I am on the right track and if so can you see my error.Tut3 Q5_2.pdf (0,5 MB)

My thought process was to diagonalize the probability matrix (A) into the eigan vectors (P) and the eigan values diagonal matrix (D), so that I could have [1 0 0] * P^-1 * D^t * P

Hope this makes my attempt clearer. Sorry I forgot I had done those calculations on the previous page

commented Oct 30, 2016 by (4,220 points)
Hi Abbi. It's difficult to see what approach you're taking here. Could you possibly edit your question and explain your thoughts in a sentence or two, either on your paper or in the body of the question?
commented Oct 30, 2016 by (4,010 points)

The question:

answered Oct 30, 2016 by (4,010 points)

I apologise, the solution is a bit messy.

I used induction to complete the proof. The letters A, B and C in my solution represent the constants in the question.

answered Oct 30, 2016 by (4,220 points)

I see what you mean now. Of course it would work to just raise the TPM to the $$t$$-th power. But it's a very difficult method. You first have to find the eigenvalues (which appear in the question as $$\lambda_1$$ and $$\lambda_2$$, but you're not told that they're eigenvalues so you would have to show this), and then you have to find the eigenvectors, and then do some matrix multiplication... you would get the answer.

But I think an inductive proof is way easier.

To get you started:

1. Find out $$p_{0,1}(1)$$ and $$p_{0,2}(1)$$ by multiplying $$(1 \ \ 0 \ \ 0 )$$ and the TPM, and checking that the formula is correct for $$t = 1$$

2. Then assume the formula correctly gives  $$p_{0,1}(t)$$ and $$p_{0,2}(t)$$. Compute $$p_{0,1}(t+1)$$ and $$p_{0,2}(t+1)$$ by multiplying the vector $$(p_{0,1}(t) \ \ \ p_{0,2}(t) \ \ \ [1 - p_{0,1}(t) - p_{0,2}(t)])$$ with the TPM. Check that it matches the given formula.

3. Conclude that the formula is true for all $$t$$.

commented Oct 30, 2016 by (640 points)

Thank You

I realize now that the induction method would have been far easier. What you explained is what I attempted but I must have made an error somewhere.