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

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

For a 2x2 matrix : $$M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

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

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

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

For a 3x3 matrix : $$M = \begin{bmatrix} a & b & c \\d & e & f \\ g & h & i \end{bmatrix}$$

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