In the book by Arora and Barak,Computational Complexity,on page 168,1st paragraph, there is a notation which I do not understand. They write For every $n \times n$ matrix $A$,and $i\in [n]$,we define $D_A(i)$ to be the $(n-1)\times (n-1)$ matrix $A_{1,i}$, i.e. A with first row and $i$th column deleted.If $x\in F\setminus[n]$;then we define $D_A(x)$ in the unique way such that for every $j,k\in [n-1]$,the function $(D_A(x))_{j,k}$ is a univariate polynomial of degree at most $n$.
My question is how this univariate polynomial is created, and how uniqueness determines its form.Is this a polynomial in $x$? What is $D_A(x)$?
OK, I think I got it. First $D_A(i)$ is defined (as a matrix) for $n$ particular values in your field $K=\Bbb Z/p\Bbb Z$. Then for every matrix position, there is a unique univariate polynomial $P\in K[x]$ of degree strictly less than $n$ (your book is off by one here) such that the corresponding entry of $D_A(i)$ equals the evaluation $P[x:=i]$ (this is a standard polynomial interpoliation result). Then define $D_A\in M_{n-1}(K[x])$ (a matrix with polynomial entries) to have these (interpolation) polynomials as entries (in their respective matrix positions).
Your book is very sloppy in using the same notation $A_{i,j}$ to denote in the first sentence a matrix obtained by dropping row$~i$ and column$~j$ from $A$, and then in the last sentence (with $D_A(x)$ as matrix) to denote simply the $i,j$ entry of the matrix.