# Thread: Applications of eigenvectors word problem

1. ## Applications of eigenvectors word problem

A radioactive isotope A decays at the rate of 2% per century into a second radioactive isotope B, which in turn decays at a rate of 1% per century into a stable isotope C.

a) Find a system of linear differential equations to describe the decay process. If we start with pure A, what are the proportions of A, B and C after 500 years, after 1,000 years and after 1,000,000 years?

This is what I've done so far:

$x_A(k)=\textrm{proportion of A after k centuries}$
$x_B(k)=\textrm{proportion of B after k centuries}$
$x_C(k)=\textrm{proportion of C after k centuries}$

$x_A(k+1)=0.98x_A(k)$

$x_B(k+1)=0.02x_A(k)+0.99x_B(k)$

$x_C(k+1)=0.01x_B(k)+x_C(k)$

$\mathbf{x}(k+1)=\begin{pmatrix}0.98&0&0\\0.02&0.99 &0\\0&0.01&1\end{pmatrix}\mathbf{x}(k)$

Let $A=\begin{pmatrix}0.98&0&0\\0.02&0.99&0\\0&0.01&1\e nd{pmatrix}$

$\rightarrow \begin{pmatrix}0.98&0&0\\0&0.99&0\\0&0&1\end{pmatr ix}$

I'm not too sure if I'm correct, I think the eigenvalues are $0.98, 0.99, 1$

At 500 years:

$\mathbf{x}(5)=A^5 \mathbf{x}(0)$

Is what I'm doing correct so far?

2. Differential equations? Or difference equations? Your approach is rather like a Markov chain; it could well be a valid approach, but I don't see any derivatives in there. In any case, how did you get this line:

$\rightarrow \begin{pmatrix}0.98&0&0\\0&0.99&0\\0&0&1\end{pmatr ix}?$

3. Originally Posted by Ackbeet
Differential equations? Or difference equations? Your approach is rather like a Markov chain; it could well be a valid approach, but I don't see any derivatives in there. In any case, how did you get this line:

$\rightarrow \begin{pmatrix}0.98&0&0\\0&0.99&0\\0&0&1\end{pmatr ix}?$
Just with row reduction

4. Is this what you're talking about. How would I find the corresponding eigenvectors. So far I'm only used to finding eigenvectors of 2x2 matrices

$\mathbf{y}(t)=\alpha e^{0.98t}\begin{pmatrix}?\\?\\?\end{pmatrix}+\beta e^{0.99t}\begin{pmatrix}?\\?\\?\end{pmatrix}+\gamm a e^{t}\begin{pmatrix}?\\?\\?\end{pmatrix}$

5. You can't use row reduction to diagonalize a matrix. So your candidate solution is, I'm afraid, incorrect. First, you need a differential equation model of the isotopes. What do you think is the correct model here?