Laplace expansion

Laplace expansion computes a determinant by walking along one row (or one column) and summing each entry times the determinant of the smaller matrix you get by deleting that entry's row and column, with an alternating plus/minus checkerboard of signs. It turns one n-by-n determinant into a weighted sum of n smaller (n-1)-by-(n-1) determinants — a recursion all the way down to 1x1.

Expanding along row i: det(A) = sum over j of a_{ij} * C_{ij}, where the cofactor C_{ij} = (-1)^(i+j) * M_{ij}, and the minor M_{ij} is the determinant of A with row i and column j struck out. The sign pattern (-1)^(i+j) is the checkerboard +, -, +, ...; you may expand along any single row or any single column and get the same answer.

It falls out of multilinearity, so you can prove rather than memorize it. Write the chosen row as a sum over standard basis vectors e_1 + ... weighted by its entries, then use linearity in that row to split the determinant into n pieces; each piece is one entry times a determinant with a basis vector in that row, which (after the alternating bookkeeping) is exactly the signed minor. Uniqueness confirms the result is det.

Caveat: as a numerical method it is even worse than it looks — expanding fully costs on the order of n! operations, the same wall as Leibniz. Its real uses are theoretical and small-scale: deriving the adjugate and the inverse formula, hand-computing 3x3 or 4x4 cases, and exploiting a sparse row or column where most a_{ij} are zero so most terms vanish.