  We prove First Fundamental Theorems of Coinvariant Theory for the standard
coactions of the quantum general and special linear groups on tensor products
of quantum matrix algebras. More precisely, let m,n,t be arbitrary positive
integers, let A and B be the quantum coordinate rings of the $t \times t$
general and special linear groups over an arbitrary field K, and let $C_{m,t}$
and $C_{t,n}$ denote the quantum coordinate rings of $m \times t$ and $t \times
n$ matrices over K. We first prove that the set of coinvariants for the
coaction of A on $C_{m,t} \otimes C_{t,n}$ equals the image of the natural
K-algebra map from the quantum coordinate ring of $m \times n$ matrices to
$C_{m,t} \otimes C_{t,n}$ induced by comultiplication. The set of coinvariants
for the coaction of B on $C_{m,t} \otimes C_{t,n}$ is shown to be the
subalgebra generated by the above image together with a tensor product of two
algebras generated by $t \times t$ quantum minors.
