cofactor (adjugate) matrix
The adjugate of a square matrix is the matrix of its cofactors, transposed. Its whole reason for existing is one beautiful identity: A times adj(A) equals det(A) times the identity. That single equation packages every cofactor-expansion fact at once and, when det(A) is nonzero, hands you a closed-form inverse with no row reduction at all.
Build it in two steps. First the cofactor matrix C has entries C_{ij} = (-1)^(i+j) * M_{ij}, the signed minors from Laplace expansion. Then adj(A) = C^T, the transpose of the cofactor matrix (note the swap of indices). The key relation is A * adj(A) = adj(A) * A = det(A) * I. The diagonal entries are genuine cofactor expansions of det(A); the off-diagonal entries are 'wrong-row' expansions, which always vanish — that is the whole trick.
The payoff is the inverse formula: if det(A) != 0 then A^-1 = (1/det(A)) * adj(A). It is the cleanest proof that invertibility is equivalent to det(A) != 0, and it shows entrywise how the inverse depends on A. Even when A is singular the identity still holds (it just reads A * adj(A) = 0), and adj remains useful in proofs, in Cramer's rule, and in differentiating determinants (the entries of adj appear in d det / d a_{ij}).
Caveat: as a way to actually compute an inverse, the adjugate is terrible for n larger than 3 — it requires n^2 minors, each itself a determinant. Use Gaussian elimination or LU for real work. Treasure the adjugate as a theoretical and symbolic tool, not a numerical one, and mind the transpose: forgetting C^T versus C is the classic error.
The 2x2 adjugate: swap the diagonal, negate the off-diagonal; the product gives det(A)*I.
Beware a name clash: 'adjoint' in inner-product theory means the conjugate transpose A^*, a different object. The cofactor matrix is the 'classical adjoint' or, better, the adjugate — use adj(A) to avoid confusion.