Since there are C(*m*,*k*) choices of *k* rows out of *m*, and there are C('\'n*,*k*) choices of *k* columns out of *n*, there are a total of C(*m*,*k*)C(*n*,*k*) minors of size *k*×*k''.

Especially important are the (*n*-1)×(*n*-1) minors of an *n*×*n* square matrix - these are often denoted *M*_{ij}, and are derived by removing the *i*th row and the *j*th column.

The **cofactors** of a square matrix *A* are closely related to the minors of *A*: the cofactor *C*_{ij} of *A* is defined as (-1)^{i+j} times the minor *M*_{ij} of *A*.

For example, given the matrix

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.

We will use the following notation for minors: if *A* is an *m*×*n* matrix, *I* is a subset of {1,...,*m*} with *k* elements and *J* is a subset of {1,...,*n*} with *k* elements, then we write [*A*]_{I,J} for the *k*×*k* minor of *A* that corresponds to the rows with index in *I* and the columns with index in *J*.

Both the formula for ordinary matrix multiplication and the Cauchy-Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices.
Suppose that *A* is an *m*×*n* matrix, *B* is an *n*×*p* matrix, *I* is a subset of {1,...,*m*} with *k* elements and *J* is a subset of {1,...,*p*} with *k* elements. Then

A more systematic, algebraic treatment of the minor concept is given in multilinear algebra, using the wedge product. If the columns of a matrix are wedged together *k* at a time, the *kxk* minors appear as the components of the resulting *k*-vectors.

In graph theory, the term