Adjugate matrix

In linear algebra, the adjugate of a square matrix A is the transpose of its cofactor matrix and is denoted by adj(A).^[1][2] It is also occasionally known as adjunct matrix,^[3][4] or "adjoint",^[5] though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.

The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix:

${\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {I} ,}$

where I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant.