The number of matrices in finite field

Question

How to prove this theorem ? The number of random $ m \times k$ $(m \geqslant k)$ matrices of rank $k$ over finite fields $\mathbb{Z}_p$ is $(p^m-1)(p^m-p)\ldots(p^m-p^{k-1})$.

Answer 1

First of all you should note that $m\geq k$. If not, the rank can never be $k$.

The columns of an $m\times k$ matrix of rank $k$ are $k$ linearly independent vectors in $\mathbb{Z}_p^m$.

For the first column, we can choose $p^m - 1$ vectors (every vector of $\mathbb{Z}_p^m$ except for 0).

Once we have fixed the first column, we can choose $p^m-p$ vectors for the second column (every vector of $\mathbb{Z}_p^m$ that is linearly independent of the first column; there are $p$ of those).

And so on, until you reach the last ($k$th) column.