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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?

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    $\begingroup$ You mean the determinant of the matrix with row $r$ and column $c$ removed, not the original matrix $A$. $\endgroup$ – Robert Israel Jan 23 at 20:51
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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.

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