Minor and Cofactor of an Element

The minor of an element in a matrix is the determinant of the submatrix obtained by deleting the row and column containing that element.
The cofactor of an element in a matrix is the minor of that element multiplied by (-1)^(i+j), where i and j are the row and column indices of the element.
Cofactors are used to find the inverse of a matrix and to solve systems of linear equations.
The determinant of a matrix can be calculated using the cofactor expansion method, which involves expanding along any row or column.
The determinant of a matrix is zero if and only if the matrix is singular, i.e., it has no inverse.

Minor of an Element: $$M_{ij} = \begin{vmatrix} a_{11} & \cdots & a_{1,j-1} & a_{1,j+1} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{i-1,1} & \cdots & a_{i-1,j-1} & a_{i-1,j+1} & \cdots & a_{i-1,n} \\ a_{i+1,1} & \cdots & a_{i+1,j-1} & a_{i+1,j+1} & \cdots & a_{i+1,n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & \cdots & a_{n,j-1} & a_{n,j+1} & \cdots & a_{nn} \end{vmatrix}$$
Cofactor of an Element: $$C_{ij} = (-1)^{i+j} M_{ij}$$