Minor, cofactor and adjoint matrix

1. Minor（余子式）

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns.

1.1 First Minor（一阶余子式）

If $A$ is a square matrix, then the minor of the entry in the i-th row and j-th column (also called the (i,j) minor, or a first minor) is the determinant of the submatrix formed by deleting the i-th row and j-th column. This number is often denoted $M_{ij}$.

For a matrix $B$,

$$\begin{equation} \left( \begin{matrix} B_{11}&B_{12}&B_{13}\\ B_{21}&B_{22}&B_{23}\\ B_{31}&B_{32}&B_{33} \end{matrix} \right) \end{equation}$$

then first minor for $B_{13}$ is

$$\begin{equation} M_{13}= \left| \begin{matrix} B_{21}&B_{22}\\ B_{31}&B_{32} \end{matrix} \right| \end{equation}$$

2.Cofactors（代数余子式）

Cofactors are form first minors, adding $(-1)^{i+1}$, which are often denoted $C_{ij}=(-1)^{-1}M_{ij}$. The cofactor for B is

$$\begin{equation} C_{13}= (-1)^{1+3}\left| \begin{matrix} B_{21}&B_{22}\\ B_{31}&B_{32} \end{matrix} \right| \end{equation}$$

Cofactors are useful for computing both the determinant and inverse of square matrices.

2.1. Cofactor expansion of the determinant

The determinant of $A$ (denoted $\text{det}(A)$) can be written as the sum of the cofactors of any row or column of the matrix multiplied by the entries that generated them, which is called Laplace expansion or Cofactor expansion

The cofactor expansion along the jth column gives

$$\text{det}(A)=\sum_{i=1}^n A_{ij}C_{ij}$$
The cofactor expansion along the ith row gives
$$\text{det}(A)=\sum_{j=1}^n A_{ij}C_{ij}$$

2.2 Cofactor for computing inverse of a matrix

If set cofactor matrix of matrix $A$ as

$$\begin{equation} C= \left( \begin{matrix} C_{11}&C_{12}&\cdots&C_{1n}\\ C_{21}&C_{22}&\cdots&C_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ C_{n1}&C_{n2}&\cdots&C_{nn} \end{matrix} \right) \end{equation}$$

which is called adjoint matrix（伴随矩阵）
Then the inverse of A is the transpose of the cofactor matrix times the reciprocal of the determinant of A:

$$A^{-1} = \frac{1}{\operatorname{det}( A)} C^{T}$$
proof: from the fact that
$$A C(A)^T = \text{det}(A)I$$
then
$$A^{-1} = \frac{1}{\operatorname{det}( A)} C^{T}$$