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I have a certain problem that I've managed to convert to a matrix problem. I have 3 unknown variables and the problem is defined by a 3x3 matrtix and a 3x1 vector. From the nature of the problem the rank is 2. My question is if it is possible to get the relation x/y - the first two unknowns. The specific problem is as follows: $$ \underbrace{\begin{pmatrix}0&2&1\\2&0&1\\1&3&2\end{pmatrix}}_{H} \ \underbrace{\begin{pmatrix}x\\y\\z\end{pmatrix}}_{t} =\underbrace{\begin{pmatrix}w_1\\w_2\\w_3\end{pmatrix}}_{b} $$ $\operatorname{rank}(H) = 2,\ \det(H) = 0$, $w_1,w_2,w_3$ are known scalars


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In general, I'm afraid not. You see, we have $$ H\begin{pmatrix}1\\1\\-1\end{pmatrix}=0. $$ Therefore, if $(x_0,y_0,z_0)$ is a solution to $Ht=b$, then $(x,y,z)=(x_0+k,y_0+k,z_0-k)$ is also a solution for any $k$. But the ratio $\frac{x}{y}=\frac{x_0+k}{y_0+k}$ is non-constant in general. The only exceptional case is $x_0=y_0=0$ and $(w_1,w_2,w_3)\not=0$, i.e. when $(w_1,w_2,w_3)$ is a nonzero multiple of $(1,1,2)$. Then we have $\frac xy=1$ for each nonzero solution $t$.

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$Z$ can be a free variable (since $rank(H)=2$), from the first equation you can express $Y$ in terms of $Z$ and from the second equation you can express $X$ in terms of $Z$.

Now you have a general expression for $X/Y$ for a given a $Z$ - check if this ration is dependent on $z$, maybe you would like to set a $z$ so get a specific ratio if it is dependent on it.

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