Leibniz formula

The Leibniz formula is the determinant written out term by term as a sum over all the ways to choose one entry from each row and column, with a plus or minus sign attached to each way. For an n-by-n matrix it has n! terms — six for a 3x3, twenty-four for a 4x4 — which is why it is a definition you cherish in theory and never use to compute large cases.

Formally, det(A) = sum over all permutations sigma of sign(sigma) * a_{1,sigma(1)} * a_{2,sigma(2)} * ... * a_{n,sigma(n)}. A permutation sigma is a bijection of {1, ..., n}: it picks, for each row i, the column sigma(i) you take the entry from, guaranteeing one pick per row and per column. The factor sign(sigma) is +1 or -1 according to the permutation's parity, and it is the direct fingerprint of the alternating axiom.

You can derive it, not memorize it, straight from the characterization. Write each column of A in the standard basis and expand det by multilinearity into a giant sum; the alternating property zeroes every term whose chosen columns repeat a basis vector, so only the terms indexed by permutations survive; reordering each surviving product back to standard order produces exactly the sign factor. Uniqueness guarantees this sum IS the determinant.

Caveat: n! grows ferociously, so Leibniz is computationally hopeless beyond tiny matrices (20! already exceeds 10^18). Real computation uses Gaussian elimination (an O(n^3) route via row reduction) or LU; Leibniz earns its keep as the source of proofs, the basis of the cofactor expansion, and the bridge to the sign of a permutation.