Let $m,n,r$ be positive integers with $r+1\le m\le n$ . Let $t$ be a variable and $A(t)=[a_{ij}(t)] $ be a $r\times r$ matrix with entries being polynomials in the variable $t$ namely $a_{ij}(t)=\sum_{k\ge 0} \binom{m-i}{k} \binom {n-j}{k+i-j}t^k$ , where by convention $\binom {a}{b}=0$ if $a<b$. Now consider the polynomial $h(t)=\det A(t)$.
Are there nice compact expressions for the following two quantities: $h(1)$ and $h'(1)$ ? At least can we get an expression when $m=n$ and $r=2$ ?