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.
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[1]) is the determinant of the submatrix formed by deleting the i-th row and j-th column.
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.
Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A, also called minor determinant of order k of A or, if m = n, the (n − k)th minor determinant of A (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.
Sometimes the term is used to refer to the k × k matrix obtained from A as above (by deleting m − k rows and n − k columns), but this matrix should be referred to as a (square) submatrix of A, leaving the term "minor" to refer to the determinant of this matrix.
corresponding to these choices of indexes is denoted
Also, there are two types of denotations in use in literature: by the minor associated to ordered sequences of indexes I and J, some authors[4] mean the determinant of the matrix that is formed as above, by taking the elements of the original matrix from the rows whose indexes are in I and columns whose indexes are in J, whereas some other authors mean by a minor associated to I and J the determinant of the matrix formed from the original matrix by deleting the rows in I and columns in J;[2] which notation is used should always be checked.
In this article, we use the inclusive definition of choosing the elements from rows of I and columns of J.
[5] The cofactors feature prominently in Laplace's formula for the expansion of determinants, which is a method of computing larger determinants in terms of smaller ones.
Given an n × n matrix A = (aij), the determinant of A, denoted 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.
then the cofactor expansion along the j-th column gives:
The cofactor expansion along the i-th row gives:
One can write down the inverse of an invertible matrix by computing its cofactors by using Cramer's rule, as follows.
Then the inverse of A is the transpose of the cofactor matrix times the reciprocal of the determinant of A:
where I′, J′ denote the ordered sequences of indices (the indices are in natural order of magnitude, as above) complementary to I, J, so that every index 1, ..., n appears exactly once in either I or I', but not in both (similarly for the J and J') and [A]I, J denotes the determinant of the submatrix of A formed by choosing the rows of the index set J and columns of index set J.
A simple proof can be given using wedge product.
so the sign is determined by the sums of elements in I and J.
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.
where the sum extends over all subsets K of {1, ..., n} with k elements.
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 k-th 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.
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,
where the coefficients agree with the minors computed earlier.
In some books, instead of cofactor the term adjunct is used.
Using this notation the inverse matrix is written this way:
Keep in mind that adjunct is not adjugate or adjoint.