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. 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 Mi,j. The (i, j) cofactor is obtained by multiplying the minor by . 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. So the cofactor of the (2,3) entry is 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, (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. For a matrix A as above, there are a total of minors of size k × k. The minor of order zero is often defined to be 1. For a square matrix, the zeroth minor is just the determinant of the matrix. Let and be ordered sequences (in natural order, as it is always assumed when talking about minors unless otherwise stated) of indexes, call them I and J, respectively.