So I know that the formula for the determinant of a matrix by expansion of minors is: $$\sum_{c=1}^n a_{rc}a'_{rc}$$ is the expansion of minors by a row and $$\sum_{r=1}^n a_{rc}a'_{rc}$$ is the expansion of minors by a column.
In both expressions $a'_{rc} = (-1)^{r+c}\det(A)$ where r,c and A are respectively the row, column, and matrix $A$. I'm not sure how this expression for the cofactor is derived? Where does it come from? Is there a way to intuitively understand where it comes from?
It depends on what you consider to be the primary definition of determinant. If you define it with the Leibniz formula $$ \det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}$$ where the sum is over all permutations of $[1,\ldots,n]$, then the minor expansion over row $i$ is just picking out, for each element $a_{ij}$ in that row, the terms involving that element.