Tool to compute a Cofactor matrix: a matrix composed of the determinants of its sub-matrices (minors).

Cofactor Matrix - dCode

Tag(s) : Matrix

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Tool to compute a Cofactor matrix: a matrix composed of the determinants of its sub-matrices (minors).

The **cofactor matrix** of a square matrix \( M \) is noted \( Cof(M) \). For each item in the matrix, compute the determinant of the sub-matrix \( SM \) associated. The determinant is noted \( \text{Det}(SM) \) or \( | SM | \) and is also called 'minor'. To calculate \( Cof(M) \) multiply each minor by a \( -1 \) factor according to the position in the matrix.

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

Calculation of a 2x2 **cofactor matrix** :

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

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

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

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

Calculation of a 3x3 **cofactor matrix** :

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

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

The transpose of the **cofactor matrix** (comatrix) is the adjoint matrix.

Most of the properties of the **cofactor matrix** actually concern its transpose, the transpose of the matrix of the cofactors is called adjugate matrix.

$$ A({}^t{{\rm com} A}) = ({}^t{{\rm com} A})A =\det{A} \times I_n $$

$$ A^{-1}=\frac1{\det A} \, {}^t{{\rm com} A} $$

A cofactor is calculated from the minor of the submatrix.

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

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