Modular matrix invariants under some transpose actions

Consider the special linear group of degree $2$ over an arbitrary finite field, acting on the full space of $2 \times 2$-matrices by transpose. We explicitly construct a generating set for the corresponding modular matrix invariant ring, demonstrating that this ring is a hypersurface. Using a recent result on $a$-invariants of Cohen-Macaulay algebras, we determine the Hilbert series of this invariant ring, and our method avoids seeking the generating relation. Additionally, we prove that the modular matrix invariant ring of the group of upper triangular $2 \times 2$-matrices is also a hypersurface.