Minor (linear algebra)

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 and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for computing both the determinant and inverse of square matrices.

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

To compute the minor M2,3 and the cofactor C2,3, we find the determinant of the above matrix with row 2 and column 3 removed.

The complement, Bijk...,pqr..., of a minor, Mijk...,pqr..., of a square matrix, A, is formed by the determinant of the matrix A from which all the rows (ijk...) and columns (pqr...) associated with Mijk...,pqr... have been removed. The complement of the first minor of an element aij is merely that element.[5]

One can write down the inverse of an invertible matrix by computing its cofactors by using Cramer's rule, as follows. The matrix formed by all of the cofactors of a square matrix A is called the cofactor matrix (also called the matrix of cofactors or, sometimes, comatrix):

Then the inverse of A is the transpose of the cofactor matrix times the reciprocal of the determinant of A:

The transpose of the cofactor matrix is called the adjugate matrix (also called the classical adjoint) of A.

Given an m × n matrix with real entries (or entries from any other field) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

A more systematic, algebraic treatment of minors is given in multilinear algebra, using the wedge product: the k-minors of a matrix are the entries in the kth exterior power map.

If the columns of a matrix are wedged together k at a time, the k × k minors appear as the components of the resulting k-vectors. For example, the 2 × 2 minors of the matrix

are −13 (from the first two rows), −7 (from the first and last row), and 5 (from the last two rows). Now consider the wedge product

where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is bilinear and alternating,

In some books, instead of cofactor the term adjunct is used.[7] Moreover, it is denoted as Aij and defined in the same way as cofactor:

Keep in mind that adjunct is not adjugate or adjoint. In modern terminology, the "adjoint" of a matrix most often refers to the corresponding adjoint operator.