# Cramer's rule and adjugate matrices #

The adjugate matrix is the transpose of the cofactor matrix. It is calculated with Cramer's rule, which we introduce first. The vectors returned by Cramer's rule are given by the linear map cramer, which sends a matrix A and vector b to the vector consisting of the determinant of replacing the ith column of A with b at index i (written as (A.update_column i b).det). Using Cramer's rule, we can compute for each matrix A the matrix adjugate A. The entries of the adjugate are the minors of A. Instead of defining a minor by deleting row i and column j of A, we replace the ith row of A with the jth basis vector; the resulting matrix has the same determinant but more importantly equals Cramer's rule applied to A and the jth basis vector, simplifying the subsequent proofs. We prove the adjugate behaves like det A • A⁻¹.

## Main definitions #

• Matrix.cramer A b: the vector output by Cramer's rule on A and b.
• Matrix.adjugate A: the adjugate (or classical adjoint) of the matrix A.
• https://en.wikipedia.org/wiki/Cramer's_rule#Finding_inverse_matrix