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 \(t\).