Is there any formula for sum of multiplication of non-intersecting elements in a matrix?
For example for a $3\times 3$ matrix \begin{align*} \begin{bmatrix} a_{11} &a_{12} &a_{13}\\ a_{21} &a_{22} &a_{23}\\ a_{31} &a_{32} &a_{33} \end{bmatrix}, \end{align*} I am looking for
$a_{11}\Big(a_{22}a_{33}+a_{23}a_{32}\Big)+a_{12}\Big(a_{21}a_{33}+a_{23}a_{31}\Big)+a_{13}\big( a_{21}a_{32}+a_{22}a_{31}\Big).$
This looks like the determinant formula except that it doesn't change sign.