# Where does the cofactor expression in the expansion of minors formula come from?

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?

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

## 1 Answer

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.