Adjoint Matrix

To calculate the adjoint matrix \( {\rm Adj} \) of the square matrix \( M \), compute \( ^{\operatorname t}{\rm Cof} \) the transpose of the cofactors matrix of \( M \).

$$ {\rm Adj}=^{\operatorname t}{\rm Cof} $$

To calculate the cofactors matrix \( {\rm Cof}(M) \), 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.

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

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

Formula for a 2x2 matrix:

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

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

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

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

Formula for a 3x3 matrix:

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

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

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

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