Manuals/calci/ADJ

ADJ(Array)

• $Array$ is the set of values.

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:$adj(A)=C^{T}={\begin{pmatrix}+{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{12}&a_{13}\\a_{32}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{12}&a_{13}\\a_{22}&a_{23}\end{vmatrix}}\\+{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{13}\\a_{31}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{13}\\a_{21}&a_{23}\end{vmatrix}}\\+{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{12}\\a_{31}&a_{32}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}\\\end{pmatrix}}$

Examples

1.adj([[10,12],[-14,21]])

Spreadsheet

21	-12
14	10

2.adj([[4,5,8],[3,-6,-9],[10,-12,4]])

Spreadsheet

-132	-116	2.9999999999999982
-102	-64	60
24	98	-39

3.adj([[5,-2,2,7],[1,0,0,3], [-3,1,5,0], [3,-1,-9,4]])

Spreadsheet

-12	76	-60	-36
-56	207.99999999999997	-81.99999999999999	-57.99999999999999
4	3.999999999999999	-1.9999999999999998	-10
4	3.9999999999999982	20	12

See Also

• MATRIXADJOINT
• MINVERSE
• MMULT

References

• Adjugate matrix