So I understand from the definition of the determinant that: $$\det(A)=\sum_{i}^{} (-1)^{i+k}a_{ki}M_{ki}$$where we define $M_{ki}$ to be the determinant of an $(n-1) \times (n-1$) matrix formed by removing the $k$-th row and $i$-th column from $A$.
How can we show that, in general, any choice of $k$ will yield exactly the same determinant. Also how can we show that the determinant can be found by expanding over a column rather than a row?