I have the system of simultaneous equations: $$ \begin{cases} y_1 = b_{12}y_2 + b_{13}y_3 + a_{11}x_{12} + a_{13}x_3 \\ y_2 = b_{21}y_1 + a_{21}x_1 + a_{22}x_{2} \\ y_3 = b_{32}y_2 + a_{31}x_1 + a_{32}x_2 + a_{33}x_3 \end{cases} $$. For the second and third equations, the necessary condition of idenfication or over-idenfication is hold true, but what about the suffiency one? As I know, for this it is necessary that the matrix determinant composed of coefficients for variables that are not present in the equation mustn't equal to zero. But the matrixs are not square. How to verify the equation for (over-)-idenfication in this case?
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Please explain what you mean by "the matrices are not square," because the equations involve two obviously square matrices $a_{ij}$ and $b_{ij}.$ – whuber May 13 '20 at 14:31
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I mean the matrix of the cofficients of $$y_3$$, $$x_{12}$$ and $$x_3$$ for the second equation. – VanyaTihonov May 13 '20 at 14:49
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You can write these equations in the form $By = Ax$ where $B$ and $A$ are both (square) $3\times 3$ matrices. If $B$ is invertible, this can be further written as $y=Cx$ where $C=B^{-1}A.$ – whuber May 13 '20 at 16:57