Minor (linear algebra)

:This article is about a concept in linear algebra. For the unrelated concept of "minor" in graph theory, see minor (graph theory).

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In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. Suppose A is an m×n matrix and k is a positive integer not larger than m and n. A k×k minor of A is the determinant of a k×k matrix obtained from A by deleting m-k rows and n-k columns.

Related Topics:
Linear algebra - Matrix - Determinant

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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.

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Especially important are the (n-1)×(n-1) minors of an n×n square matrix - these are often denoted Mij, and are derived by removing the ith row and the jth column.

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The cofactors of a square matrix A are closely related to the minors of A: the cofactor Cij of A is defined as (−1)i+j times the minor Mij of A.

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For example, given the matrix

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:egin{pmatrix}

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1 & 4 & 7 \

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3 & 0 & 5 \

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