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Cofactor Matrix

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

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Cofactor Matrix -

Tag(s) : Mathematics,Algebra,Symbolic Computation

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# Cofactor Matrix

## Cofactor Matrix Calculator

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

### How to calculate the matrix of cofactors?

The cofactor matrix is computed from the square matrix M. For each item in the matrix, one computes the determinant of the sum matrix SM associated (also called Minor) with a -1 factor according to the position in the matrix$${\rm Cof}_{i,j}=(-1)^{i+j}\text{Det}(SM_i)$$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 : $$M = \begin{pmatrix} a & b & c \\d & e & f \\ g & h & i \end{pmatrix}$$$${\rm Cof}(M) = \begin{pmatrix} +\begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} d & e \\ g & h \end{vmatrix} \\ & & \\ -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} \\ & & \\ +\begin{vmatrix} b & c \\ e & f \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{pmatrix}$$