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Cofactor Matrix

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

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Cofactor Matrix -

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# Cofactor Matrix

## Matrix Cofactor Calculator

### What is the matrix of cofactors? (Definition)

The cofactor matrix of a square matrix $M = [a_{i,j}]$ is noted $Cof(M)$. It is the matrix of the cofactors, i.e. the minors weighted by a factor $(-1)^{i+j}$.

### How to calculate the matrix of cofactors?

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.

### What are the properties of the cofactor 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.

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

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

### What is the cofactor of a matrix?

A cofactor is calculated from the minor of the submatrix.

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

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