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?

  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.