I want to find an $n\times n$ matrix $A$ such that $\det{A}\neq 0$ for any $a_{11}$. I am working in the field of complex numbers. I know that a matrix does not have a zero determinant if its rows/columns are independent. But if I vary $a_{11}$ couldn't I always make a matrix that is dependent?
To be more straightforward, I am looking for a matrix, $$A=\begin{bmatrix}a_{11}&b\\c&d\end{bmatrix},$$ where $b,c,d$ are fixed, and $a_{11}$ is allowed to vary over all real numbers, such that the determinant is never $0$. Howe could I go about finding such a matrix?
Hint: Knowing that interchanging two rows results in the determinant being multiplied with $(-1)$, can you mabye think of a matrix where regardless of the entry $a_{n,1}$ the determinant of this matrix is not equal to zero?