Minor (linear algebra)

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition.

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_{i, j}. The (i, j) cofactor is obtained by multiplying the minor by (−1)^{i + j}.

To illustrate these definitions, consider the following 3 × 3 matrix,

$${\displaystyle {\begin{bmatrix}1&4&7\\3&0&5\\-1&9&11\\\end{bmatrix}}}$$

To compute the minor M_2,3 and the cofactor C_2,3, we find the determinant of the above matrix with row 2 and column 3 removed.

$${\displaystyle M_{2,3}=\det {\begin{bmatrix}1&4&\Box \\\Box &\Box &\Box \\-1&9&\Box \\\end{bmatrix}}=\det {\begin{bmatrix}1&4\\-1&9\\\end{bmatrix}}=9-(-4)=13}$$

So the cofactor of the (2,3) entry is

$${\displaystyle C_{2,3}=(-1)^{2+3}(M_{2,3})=-13.}$$