Laplace expansion along a row

Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$. Let $C_{ij}$ be the cofactors of $A$. Then, for each row $i$, $$\det A = \sum_{j=1}^{n} a_{ij} C_{ij} = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$

Remarks

The expansion reduces a determinant of order $n$ to $n$ determinants of order $n - 1$. It is efficient when the chosen row contains many zeros. There is a similar expansion along a column. Replace the entries of row $i$ by those of another row $k$. Then the sum is $0$, because the new matrix has two equal rows.

Explanations

L1 human

To expand a determinant along a row, take each entry of that row. Multiply it by its cofactor, which is the signed determinant of what is left after deleting its row and column. Add these products.