I have a coupled system of three equations that I am trying to solve via matrices and I am having trouble figuring out how to write out my matrices. My three equations are as follows:
$-sx+sy=0$
$rx-y-xz=0$
$xy-bz=0$
where s, r, and b are given parameters and $x, y, z$ are my three variables. My trouble is the "$xz$" and the "$xy$" terms since they are coupled between two different variables. Ignoring those two terms, my matrix equation would read:
\begin{align}
\begin{bmatrix}-s & s & 0 \\ r & -1 & ? \\ 0 & ? & -b \end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}=\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}
\end{align}
Where "$?$" represents the coupled terms. For the life of me I cannot figure out how I need to write those two terms in my matrix equation to allow me to solve this equation. I first thought that I could break my $3 \times3$ matrix into two separate $3 \times 3$ matrices that when combined, gave me what I wanted but after some scribbling, that came up short. Do any of you know what I can do to finish writing my matrix?
Best Answer
(1) $-sx+sy=0$
(2) $rx-y-xz=0$
(3) $xy-bz=0$
You can try something like this:
(1)$\rightarrow x=y$
(3)$\rightarrow z=\frac{x^2}{b}$
(2)$\rightarrow xz=rx-x$
And continue in the similar manner.