# How would you solve this $9$ system of equations with multi variables?

Question: Solve for $\alpha,\beta,\gamma,\delta,\theta,\lambda,\upsilon,\phi,\omega$ in the following:$$\begin{cases}\alpha r+\beta u+\gamma x=1\\\alpha s+\beta v+\gamma y=0\\ \alpha t+\beta w+\gamma z=0\\\delta r+\theta u+\lambda x=0\\\delta s+\theta v+\lambda y=1\\\delta t+\theta w+\lambda z=0\\\upsilon r+\phi u+\omega x=0\\\upsilon s+\phi v+\omega y=0\\\upsilon t+\phi w+\omega z=1\tag1\end{cases}$$

I didn't really try much due to the sheer ugliness of the system and how intimidating it looks. I want the $\alpha,\beta,\gamma,\delta,\theta,\lambda,\upsilon,\phi,\omega$ to be in "matrix" form. Meaning if $\alpha$ is expressible as $ad-bc$, instead, write $\begin{bmatrix}a & b\\c & d\end{bmatrix}$. I have provided an example below:

Example: Find $\alpha,\beta,\gamma,\lambda$ from the following:$$\begin{cases}\alpha w+\beta y=1\\\alpha x+\beta z=0\\\gamma w+\lambda y=0\\\gamma x+\lambda z=1\tag2\end{cases}$$

I have found the solutions as \begin{align*} & \alpha=\frac z{wz-xy}=\frac z{\begin{bmatrix}w & x\\y & z\end{bmatrix}}\\ & \beta=\frac b{xy-wz}=\frac x{\begin{bmatrix}x & w\\z & y\end{bmatrix}}\\ & \gamma=\frac y{xy-wz}=\frac y{\begin{bmatrix}x & w\\z & y\end{bmatrix}}\\ & \lambda=\frac w{wz-xy}=\frac w{\begin{bmatrix}w & x\\y & z\end{bmatrix}}\end{align*}\tag3

From $(3)$, notice how instead of writing the denominator of each of the unknown as $wz-xy$ or $xy-wz$, I write it as a matrix $\begin{bmatrix}w & x\\y & z\end{bmatrix}$ and $\begin{bmatrix}x & w\\z & y\end{bmatrix}$ respectively.

Note: I know that this is probably one of the more "weirder" type of questions, but just try you're best..? I have no more attempts after this.

• Can you inspire somewhat more by giving the context in which you came across this beast? – Qwerty Nov 2 '16 at 5:46
• Check if you have $v$ the same as $\nu$. (Does $v$ appear twice in the 8th equation?) – Andrei Nov 2 '16 at 5:52
• @Qwerty Yeah sure! :) $$\begin{bmatrix}\alpha & \beta & \gamma\\\delta & \theta & \lambda\\\upsilon & \phi & \omega\end{bmatrix}\cdot\begin{bmatrix}r & s & t\\ u & v & w\\ x & y & z \end{bmatrix}=\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$$ – Frank Nov 2 '16 at 5:53
• @Andrei Actually, the first one is upsilon (just happens to look like v... Should probably change that). And the other one is the normal $v$ from the English alphabet. There's also \omega, and $w$. (Note the second one is from the English alphabet) – Frank Nov 2 '16 at 5:54
• @Frank Sure..A wonderful job.. Just you need to make sure that the wheel is circular and smooth. Don't make improvements on a rugged wheel.. :-P – Qwerty Dec 29 '16 at 20:52