Search for a tool
Cofactor Matrix

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

Results

Cofactor Matrix -

Tag(s) : Mathematics, Algebra, Symbolic Computation

dCode and you

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day!
You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? Contact-me!


Team dCode read all messages and answer them if you leave an email (not published). It is thanks to you that dCode has the best Cofactor Matrix tool. Thank you.

Cofactor Matrix

Sponsored ads

Matrix Cofactor Calculator

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

Answers to Questions

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, compute the determinant of the associated sub-matrix \( SM \). The determinant is noted \( \text{Det}(SM) \) or \( | SM | \) and is also called minor. To calculate \( Cof(M) \) 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 is the adjoint matrixhref.

What are the properties of the cofactor matrix?

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

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

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

What is the cofactor of a matrix?

A cofactor is calculated from the minor of the submatrix.

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

Ask a new question

Source code

dCode retains ownership of the source code of the script Cofactor Matrix. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, etc.) which dCode owns rights can be transferred after sales quote. So if you need to download the Cofactor Matrix script for offline use, for you, your company or association, see you on contact page !

Questions / Comments


Team dCode read all messages and answer them if you leave an email (not published). It is thanks to you that dCode has the best Cofactor Matrix tool. Thank you.


Source : http://www.dcode.fr/cofactor-matrix
© 2017 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode