Consider a finite field $\mathbb{F}_q$ of characteristic $p>0$. Let $A=(a_{ij})$ be an integer matrix with $k$ columns and a finite number of rows. Consider the algebraic subgroup $\pmb{H}_A$ of the torus $\pmb{G}_m^k$, defined on $\mathbb{F}_q$-points as $$H_A(\mathbb{F}_q)=\{(x_j)\in(\mathbb{F}_q^\times)^k\,:\,\prod_{j=1}^kx_j^{a_{ij}}=1\text{ for all }i\}.$$
I am interested in computing the number of connected components of $\pmb{H}_A$, which I expect to involve the gcd of all entries of $A$. My strategy so far has been to see $\pmb{H}_A$ as a diagonalizable algebraic group, being an algebraic subgroup of the torus $\pmb{G}_m^k$. Therefore, one can write $\pmb{H}_A$ as a product of copies of $\pmb{G}_m$ and copies of $\pmb{\mu}_n$ for certain integer values $n\geq1$ (Proposition 12.3 in Milne's Algebraic groups). However, I'm unsure what this product might look like and how I can deduce from it the number of connected components of $\pmb{H}_A$.