Cramer's rule

Cramer's rule gives each unknown in a square linear system as a ratio of two determinants. To find x_k, take det(A) for the denominator, then for the numerator replace the k-th column of A with the right-hand side b and take that determinant. Divide. It is a strikingly explicit formula — every solution coordinate written purely in determinants of the data.

Precisely, for A x = b with A invertible (det(A) != 0), the solution is x_k = det(A_k) / det(A), where A_k is A with its k-th column replaced by b. The proof is pure multilinearity: det(A_k) is the determinant of A with column k swapped for b = x_1*(col 1) + ... + x_n*(col n); expand by linearity in that column, and the alternating property kills every term except the one carrying x_k times det(A).

Its value is conceptual and theoretical. Cramer's rule proves the solution exists and is unique exactly when det(A) != 0, and it shows how each x_k depends smoothly (rationally) on the entries of A and b — invaluable for sensitivity analysis, symbolic answers, and existence proofs. It also makes the link to the adjugate transparent: it is just the inverse formula A^-1 = adj(A)/det(A) read column by column.

Caveat: do NOT compute with it. For n unknowns it asks for n+1 determinants, each itself expensive, giving wildly more work than Gaussian elimination's O(n^3) — and it is numerically fragile near singular A, where the ratio of two near-zero determinants is unstable. Cramer's rule is for understanding and for tiny or symbolic systems, not for production solving.