Ok I recalculated and got
[-(4/5)*2^n+ (9/5)*.5^n (6/5)*2^n - (1/5)*.5^n
-(4/5)*3^n- (9/5)*.5^n (6/5)*3^n + (1/5)*.5^n]
for the product of the three matricies
Closer, but still no cigar. I think you're making arithmetic errors in your matrix multiplication. You've got
$\displaystyle P=\begin{bmatrix}
2 &-1\\
3 &1
\end{bmatrix},\quad D^{n}=\begin{bmatrix}
1 &0\\
0 &(-0.5)^{n}
\end{bmatrix},\quad\text{and}\quad P^{-1}=\begin{bmatrix}
\frac{1}{5} &\frac{1}{5}\\
-\frac{3}{5} &\frac{2}{5}
\end{bmatrix}.$
Now, one step at a time. Show me either $\displaystyle PD^{n}$ or $\displaystyle D^{n}P^{-1}.$ Then show me $\displaystyle PD^{n}P^{-1}.$
$\displaystyle PD^{n}=\begin{bmatrix}
2 &(0.5)^{n}\\
3 &(-0.5)^{n}
\end{bmatrix},\quad\text{and}\quad PD^{n}P^{-1}=\begin{bmatrix}
\frac{2}{5}-\frac{3}{5}(0.5)^{n} &\frac{2}{5}+\frac{2}{5}(0.5)^{n}\\
\frac{3}{5}+\frac{3}{5}(0.5)^{n} &\frac{3}{5}-\frac{2}{5}(0.5)^{n}
\end{bmatrix}.$
I recalculated, is it correct this time?
I get
$\displaystyle PD^{n}=
\begin{bmatrix}
2 &-(-0.5)^{n}\\
3 &(-0.5)^{n}
\end{bmatrix},$ and
$\displaystyle PD^{n}P^{-1}=\begin{bmatrix}
0.6(-0.5)^{n}+0.4 &-0.4(-0.5)^{n}+0.4\\
-0.6(-0.5)^{n}+0.6 &0.4(-0.5)^{n}+0.6
\end{bmatrix}.$
You're even closer than before. Just watch those minus signs!
So what happens next?
Well, you know that $\displaystyle x_{1}+x_{2}=1.$ Therefore, $\displaystyle x_{2}=1-x_{1}.$ Hence, I can represent the $\displaystyle x$ vector as
$\displaystyle x=\begin{bmatrix}x_{1}\\ 1-x_{1}\end{bmatrix}.$
Multiply your $\displaystyle PD^{n}P^{-1}$ times this vector, and then take the limit. What do you get here?
$\displaystyle PD^{n}P^{-1}X=\begin{bmatrix}
(0.6(-0.5)^{n}+0.4)x_{1} + (1-x_{1})(-0.4(-0.5)^{n}+0.4)\\
(-0.6(-0.5)^{n}+0.6)x_{1} + (1-x_{1})(0.4(-0.5)^{n}+0.6)
\end{bmatrix}.$
taking the limit, wouldn't all the ^n terms go to 0? So then I have
$\displaystyle \begin{bmatrix}
(0.4)x_{1} + (1-x_{1})(0.4)\\
(0.6)x_{1} + (1-x_{1})(0.6)
\end{bmatrix}.$
so the terms cancel to
$\displaystyle \begin{bmatrix}
0.4\\
0.6
\end{bmatrix}.$
Excellent! In your write-up, you might want to mention why all the exponentiated terms vanish in the limit.
So, that pretty much wraps up your first two problems. Have you gotten a good start on the third problem?
Just as a heads-up: it is considered better both to write up your original problem in LaTeX, as well as not to include more than two related problems in one thread.
I enjoyed these problems myself. Have a good one!
Thank you so much for guiding me through it, it helps so much more than just giving me an answer.
I'm trying to do the third but I can't find a relationship between eigenvectors of A and A transpose. The eigenvalues are the same but that doesn't mean it will have all the eigenvectors unless its guaranteed you get an eigenvector when you have an eigenvalue, which i'm pretty sure it's not.
Thanks for the heads-up, I'll take the time to write it up next time, it just takes a while since I'm not too familiar with the LaTeX