Let $A$ be a $n\times m$ random matrix with entries from finite field $\mathbb{F}_q$. What is the probability that rank of $A$ is full.

Question

I know that complex-valued random matrix say $\mathbf{A} \in \mathbb{C}^{M \times N}$ with size $M \times N$ has full-rank, here is the proof

Can some please give me some direction regarding this? Is this result true.

It is important that you specify the distribution of the matrix. If for instance $A$ is the $0$ matrix with probability $1$, then it's rank is $0$ with probability $1$. Maybe you mean to have the entries drawn i.i.d. from a uniform distribution over the field ? — P. Quinton, Feb 26 '21 at 15:39
Here is the answer for the $n\times n$ case: https://math.stackexchange.com/questions/1399406/what-is-the-number-of-invertible-n-times-n-matrices-in-operatornamegl-nf. It is not to hard to modify to general $m\times n$. — Mike Earnest, Feb 26 '21 at 15:45
@MikeEarnest Thank you very much. How can we modify the result for $n\times m$. Can you please give some hint> — Shweta Aggrawal, Feb 28 '21 at 05:55
@P.Quinton You are right. I meant that distribution is uniform. Is the result I am seeking true? — Shweta Aggrawal, Feb 28 '21 at 05:57
To modify the result to $n\times m$ you need to ask yourself how many way there is to make $a_i$ not in the span of ${ a_1,\dots,a_{i-1}}$ given that they are linearly independent. I recommend you write a fully detail computation as an answer to your question so that people can review it. — P. Quinton, Feb 28 '21 at 08:21

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