How to find $|\{A \in \mathbb F _7^{5 \times 5}|A \text{ is invertible}\}|$?

Question

How to find $|\{A \in \mathbb F _7^{5 \times 5}|A \text{ is invertible}\}|$?

$$ \begin{align} GL_n(K) & = \{ A \in K^{n \times n }| A \text{ is invertible}\} \\ & = \{ A \in K^{n \times n }| \phi_A \text{ is an isomorphism}\} \\ & = \{ A \in K^{n \times n }| rk_{\phi A} = \text{dim }K^{n \times n}\} \\ & = \{ A \in K^{n \times n }| rk A = n\} \end{align} $$

And also

$Col(A) = Im \phi A$

$rk_KA = dim_kCol(A)$

Question: How can I zip this information to get the cardinality of all possible invertible matrices in $\mathbb F _7^{5 \times 5}$?

This post seems equivalent but somehow it does not help me. A technique without determinants would be the best for me at the moment. I appreciate also solutions or help using determinants.

Answer 1

The columns must form a basis of $\Bbb F_7^5$. For the $k$th column, you can pick any of $7^5$ vectors, except those $7^{k-1}$ that are in the $(k-1)$-dimensional subspace generated by the preceding $k-1$ columns. So the count is $$(7^5-7^0)(7^5-7^1)(7^5-7^2)(7^5-7^3)(7^5-7^4) $$