Answer
False
Work Step by Step
Let $A=\begin{bmatrix}
1 & 1&1\\1&1&1\\1&1&1
\end{bmatrix}$ be a $3×3$ matrix whose entries are all nonzero
We can see that $\det(A)=0$
Replace the entry $a_{ij}$ in $A$ with any real number $x$, we will get the matrix $A_{ij}(x)$, which still has two equal rows. As a result, $\det(A_{ij}(x))=0$