Adjoint Matrix

A square matrix $ M $ has for adjugate/adjoint matrix $ \operatorname{Adj}(M) = ^{\operatorname{t}}\operatorname{Cof}(M) $ that is the transpose of the cofactors matrix of $ M $.

The adjoint matrix $ \operatorname{Adj} $ of the square matrix $ M $ is computed $ ^{\operatorname t}\operatorname{Cof} $ as the transpose of the cofactors matrix of $ M $.

To calculate the cofactors matrix $ \operatorname{Cof}(M) $, compute, for each value of the matrix in position $ (i,j) $, the determinant of the associated sub-matrix $ SM $ (called minor) and multiply with a $ -1 $ factor depending on the position in the matrix.

$$ \operatorname{Cof}_{i,j} = (-1)^{i+j}\operatorname{Det}(SM_i) $$

To get the adjoint matrix, take the transposed matrix of the calculated cofactor matrix.

Formula for a 2x2 matrix:

$$ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

$$ \operatorname{Cof}(M) = \begin{bmatrix} {{d}} & {{-c}} \\ {{-b}} & {{a}} \end{bmatrix} $$

$$ \operatorname{Adj}(M) = \begin{bmatrix} {{d}} & {{-b}} \\ {{-c}} & {{a}} \end{bmatrix} $$

Example: $$ M = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} \Rightarrow \operatorname{Cof}(M) = \begin{bmatrix} {{1}} & {{-2}} \\ {{-3}} & {{4}} \end{bmatrix} \Rightarrow \operatorname{Adj}(M) = \begin{bmatrix} {{1}} & {{-3}} \\ {{-2}} & {{4}} \end{bmatrix} $$

Formula for a 3x3 matrix:

$$ M = \begin{bmatrix} a & b & c \\d & e & f \\ g & h & i \end{bmatrix} $$

$$ \operatorname{Cof}(M) = \begin{bmatrix} +\begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} d & e \\ g & h \end{vmatrix} \\ & & \\ -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} \\ & & \\ +\begin{vmatrix} b & c \\ e & f \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{bmatrix} $$

$$ \operatorname{Adj}(M) = \begin{bmatrix} +\begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} b & c \\ e & f \end{vmatrix} \\ & & \\ -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} \\ & & \\ +\begin{vmatrix} d & e \\ g & h \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{bmatrix} $$

Adjugate matrix, adjoint matrix or adjunct matrix are the same.