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What is an adjugate matrix? (Definition)

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 $M$.

How to compute the adjugate matrix?

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:

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

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

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

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