Adjugate matrix
The adjoint, classical adjoint (not to be confused with the real adjoint matrix ) or complementary matrix of a matrix is a term from the mathematical subfield of linear algebra. It refers to the transpose of the cofactor matrix, ie the transpose of that matrix whose entries are the signed minors ( minors ).
With the help of co-factors, one can calculate the inverse of a regular square matrix.
Definition
The adjoint of a square matrix with entries from a body is defined as
It should be noted in this case that is at the site of the cofactor. The cofactors are calculated as follows
The minors are thus the value of the determinant of the matrix caused by the strike of the - th row and the -th column.
Since the cofactor in today's textbooks rarely shows up and in older plants, the notation is not always clear, caution is advised. Often the same notation for the adjoint and the adjoint is ( ie with real matrices whose transpose, complex conjugate transpose of matrices whose ) was used.
Examples
(2 × 2 ) matrix
Any matrix in the form of
The adjoint of this matrix
(3 x 3 ) matrix
Any matrix in the form of
The adjoint of this matrix
Properties
The following relations hold for all matrices from
For additional invertible matrices applies
Calculation of the inverse of a matrix
Main article: Regular Matrix
The individual columns of the inverse of a matrix are each formed by the solution of the system with the - th unit vector on the right side. If we calculate this with the Cramer 's rule, we obtain the formula
Any matrix can thus invert a very simple way: