can someone please help me with the same ?

the main point i was trying to suggest was that the size of each complex entry has an upper bound.

its a rough idea. One way is to have 2 bits for sign (for each real and imaginary part), and 2 binary strings of some fixed size (one for complex another for real). Plus 2 more position counters of fixed size, that tell the position of decimal in each real and imaginary part respectively.

@MichaelEngelhardt Size of a vector - vector length (updating). Size of an entry - number of bits or digits (I couldn't figure out an accurate word for this)

thank you. much clearer..

thanks again. Is there a small non trivial example (soluble modulo all primes) to make things clearer that satisfies these conditions, namely: (1) subset of rows don't add up to 1 vector (2) Ranks of $M$ and $M'$ are equal (3) Radicals of of $kth$ determinant divisors of $M$ and $M′$ are the same for each $k$?

updated........

And a follow up query: Is there a scenario where: Given a matrix M such that there are no subset of rows which add up to all 1 rows. But, $xM=ιK,$ can be solved for all primes $X$? Can we have a test or property that characterizes such matrices completely?

Thank you very much. Few clarifications: As I understand the R-C theorem provides a test whenever the system will have a solution. I am unclear about the second "Let us assume.." part: Does this test of insolubility works only when $M$ has a full rank and fails otherwise when $M$ does not have a full rank? Or the insoluble scenario arises iff $M$ has a full rank and thus this test is self-sufficient for all scenarios?

can someone please help with this ?

@MaxAlekseyev Thank you. Please consider the general case then (since both are equivalent)