adjugate matrix

(Note that the first and last column of are equal to those of the adjugate matrix of ).

The transpose of the cofactor matrix is called the adjugate matrix (also called the classical adjoint) of A.

For example, calculating the inverse of a matrix via Laplace's formula (Adj (A) denotes the adjugate matrix of A)

Writing the transpose of the matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices, but this recursive method is inefficient for large matrices.

Since A is an arbitrary square matrix, this proves that adj(A) can always be expressed as a polynomial in A (with coefficients that depend on A), something that is not obvious from the definition of the adjugate matrix.

The only difference with commutative case is that one should pay attention that all determinants are calculated as column-determinants and also adjugate matrix stands on the right, while commutative inverse to the determinant of "M" stands on the left, i.e. due to non-commutativity the order is important. "