Tool to compute an Adjoint Matrix for a square matrix. Adjoint/Adjugate/Adjacency Matrix is name given to the transpose of the cofactors matrix.
Adjoint Matrix - dCode
Tag(s) : Matrix
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Adjoint Matrix
Adjugate Matrix Calculator (NxN)
Answers to Questions (FAQ)
What is an adjugate matrix? (Definition)
A square matrix $ M $ has for adjugate/adjoint matrix $ \operatorname{Adj}(M) = ^{\operatorname{t}}\operatorname{Cof}(M) $ that is the transpose of the cofactors matrix of $ M $.
How to compute the adjugate matrix?
The adjoint matrix $ \operatorname{Adj} $ of the square matrix $ M $ is computed $ ^{\operatorname t}\operatorname{Cof} $ as the transpose of the cofactors matrix of $ M $.
To calculate the cofactors matrix $ \operatorname{Cof}(M) $, compute, for each value of the matrix in position $ (i,j) $, the determinant of the associated sub-matrix $ SM $ (called minor) and multiply with a $ -1 $ factor depending on the position in the matrix.
$$ \operatorname{Cof}_{i,j} = (-1)^{i+j}\operatorname{Det}(SM_i) $$
To get the adjoint matrix, take the transposed matrix of the calculated cofactor matrix.
Formula for a 2x2 matrix:
$$ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$
$$ \operatorname{Cof}(M) = \begin{bmatrix} {{d}} & {{-c}} \\ {{-b}} & {{a}} \end{bmatrix} $$
$$ \operatorname{Adj}(M) = \begin{bmatrix} {{d}} & {{-b}} \\ {{-c}} & {{a}} \end{bmatrix} $$
Example: $$ M = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} \Rightarrow \operatorname{Cof}(M) = \begin{bmatrix} {{1}} & {{-2}} \\ {{-3}} & {{4}} \end{bmatrix} \Rightarrow \operatorname{Adj}(M) = \begin{bmatrix} {{1}} & {{-3}} \\ {{-2}} & {{4}} \end{bmatrix} $$
Formula for a 3x3 matrix:
$$ M = \begin{bmatrix} a & b & c \\d & e & f \\ g & h & i \end{bmatrix} $$
$$ \operatorname{Cof}(M) = \begin{bmatrix} +\begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} d & e \\ g & h \end{vmatrix} \\ & & \\ -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} \\ & & \\ +\begin{vmatrix} b & c \\ e & f \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{bmatrix} $$
$$ \operatorname{Adj}(M) = \begin{bmatrix} +\begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} b & c \\ e & f \end{vmatrix} \\ & & \\ -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} \\ & & \\ +\begin{vmatrix} d & e \\ g & h \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{bmatrix} $$
What is the difference between the Adjugate and Adjoint Matrix?
Adjugate matrix, adjoint matrix or adjunct matrix are the same.
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Adjoint Matrix on dCode.fr [online website], retrieved on 2024-11-21,
