Adjugate Matrix / Minor / Cofactor
Table of contents
What is an adjugate matrix
The adjugate matrix of a $n \times n$ square matrix $\boldsymbol{A}$, denoted by $\operatorname{adj}(\boldsymbol{A})$ is defined as the transpose of the cofactor matrix $\boldsymbol{C}$ of $\boldsymbol{A}$.
$$ \operatorname{adj}(\boldsymbol{A}) = \boldsymbol{C}^\top $$
Every square matrix has an adjugate matrix.
Product of a matrix and its adjugate matrix
One important property of the adjugate matrix:
The product of a matrix and its adjugate matrix is a diagonal matrix whose diagonal elements are all equal to the determinant of the matrix.
$$ \boldsymbol{A} \operatorname{adj}(\boldsymbol{A}) = \operatorname{adj}(\boldsymbol{A}) \boldsymbol{A} = \det(\boldsymbol{A}) \boldsymbol{I}_n $$
Cofactor matrix
Minor
To understand the cofactor matrix, we first need to understand the minor of a matrix.
Technically, this is the first minor of a matrix.
The $(i,j)$ minor of a $n \times n$ square matrix $\boldsymbol{A}$, denoted
$$ \boldsymbol{M}_{ij} $$
is the determinant of the $(n-1) \times (n-1)$ submatrix obtained by removing the $i$-th row and $j$-th column of $\boldsymbol{A}$.
There are $n^2$ ways to choose the $i$-th row and $j$-th column of $\boldsymbol{A}$ of which to remove.
So these $n^2$ minors can be arranged in a $n \times n$ matrix $\boldsymbol{M}$.
Example
Given a $3 \times 3$ matrix $\boldsymbol{A}$,
\[\boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\]The $(3, 2)$ minor of $\boldsymbol{A}$ is:
\[\boldsymbol{M}_{32} = \det \begin{vmatrix} 1 & 3 \\ 4 & 6 \end{vmatrix} = 1 \times 6 - 3 \times 4 = -6\]Cofactor
The $(i,j)$ cofactor of a $n \times n$ square matrix $\boldsymbol{A}$ is:
$$ \boldsymbol{C}_{ij} = (-1)^{i+j} \boldsymbol{M}_{ij} $$
i.e. the minor of $\boldsymbol{A}$ multiplied by $(-1)^{i+j}$.
Example continued
Continuing from the example above, we have $\boldsymbol{M}_{32} = -6$.
Then:
\[\boldsymbol{C}_{32} = (-1)^{3+2} \boldsymbol{M}_{32} = -(-6) = 6\]