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Tool to compute an Adjoint Matrix (Adjugate) for a square matrix, name given to the transpose of the cofactors matrix.

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Tag(s) : Mathematics,Algebra,Symbolic Computation

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## Adjoint of a Matrix Calculator

Tool to compute an Adjoint Matrix (Adjugate) for a square matrix, name given to the transpose of the cofactors matrix.

### How to compute the adjugate matrix?

To compute the adjoint matrix of the square matrix M, you can compute the transpose of the cofactors matrix. For each value of the matrix, you can compute the determinant of the associated sub-matrix SM (called minor) and multiply with a -1 factor depending on the position in the matrix.$${\rm Cof}_{i,j}=(-1)^{i+j}\text{Det}(SM_i)$$$${\rm Adj}=^{\operatorname t}{\rm Com}$$For a 2x2 matrix: $$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$$${\rm Cof}(M) = \begin{pmatrix} \,\,\,{{d}} & \!\!{{-c}}\\ {{-b}} & {{a}} \end{pmatrix}$$For a 3x3 matrix: $$M = \begin{pmatrix} a & b & c \\d & e & f \\ g & h & i \end{pmatrix}$$$${\rm Cof}(M) = \begin{pmatrix} +\begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} d & e \\ g & h \end{vmatrix} \\ & & \\ -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} \\ & & \\ +\begin{vmatrix} b & c \\ e & f \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{pmatrix}$$you then can take the transposed matrix :$${\rm Adj}(M) = \begin{pmatrix} \,\,\,{{d}} & \!\!{{-b}}\\ {{-c}} & {{a}} \end{pmatrix}$$