Manuals/calci/ADJ

ADJ(Array)

Description

This function shows the Adjoint of a given matrix.
In $ADJ(Array)$ , $Array$ is the set of matrix values.
Adjoint of a matrix is called adjugate, classical adjoint, or adjunct.Adjoint of a matrix formed by taking the transpose of the cofactor matrix of a given original Square matrix.
Adjoint of matrix A is written by $adjA$ .
The adjugate of A is the transpose of the cofactor matrix C of A, $adj(A)=C^{T}$ .
Also adjoint of a matrix is defined by $adj(A)=det(A).A^{-1}$ .
The adjugate of 1x1 matrix is $I=(1)$ .
The adjugate of 2x2 matrix $:A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ is $adj(A)={\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}$ .
Consider3x3 matrix $A={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}$ .
Its adjugate is the transpose of its cofactor matrix: $adj(A)=C^{T}={\begin{pmatrix}+{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{12}&a_{13}\\a_{32}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{12}&a_{13}\\a_{22}&a_{23}\end{vmatrix}}\\+{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{13}\\a_{31}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{13}\\a_{21}&a_{23}\end{vmatrix}}\\+{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{12}\\a_{31}&a_{32}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}\\\end{pmatrix}}$