Cofactor theorem Shilov on page 12 says determinant $D=a_{1j}A_{1j}+a_{2j}A_{2j}+...+a_{nj}A_{nj}...(I)$ is an identity in the quantities $a_{1j}, a_{2j},..., a_{nj}$.  Therefore it remains valid if we replace $a_{ij} (i = 1, 2,. . . , n)$ by any other quantities.  The quantities $A_{1j}, A_{2j},..., A_{nj}$ remain unchanged when such a replacement is made, since they do not depend on the elements $a_{ij}$.
Quantity $A_{ij}$ is called the cofactor of the element $a_{ij}$ of the determinant $D$.
My question,from the equation (I) we see that all $\Large A$s are multiplied with specific $\Large a$s, not just any $\Large a$, then how did he conclude the above statement? .

The identity (I) uses the elements $a_{1j}, a_{2j} \ldots$, which of course for the matrix have specific values. What the statement means is that if those values, and only those values, are changed in the matrix, the new determinant is valid for the new matrix.
