How to judge a determinant equals non-zero in cryptography? In the paper [1,p5], the authors construct a matrix to find out $n$ solutions of a equation set as follows:



where $c_i$ is generated by the adversary, $c$ is picked by the simulator. Note that $c^i$ means different forgeries instead of exponent.

My question is, they find out $n$ solutions iff the determinant equals non-zero, but they don't prove why the determinant equals non-zero. There is no need to prove it?

(I'm not true this is a crypto problem or not, it seems more like a math problem?)

[1] Liu, J. K., Au, M. H., Susilo, W., &amp; Zhou, J. (2014). Linkable ring signature with unconditional anonymity. IEEE Transactions on Knowledge and Data Engineering, 26(1), 157–165. https://doi.org/10.1109/TKDE.2013.17
For linear system $Ax = \beta$ with a vector on $n$ unknowns $x$, $A$ is the Vandermonde matrix, produced with powers of $c_i$. For such a matrix, determinant is a product of $(c_i - c_j)$, and so is non-zero for different $c_j$ in the set of field elements.