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.

Definition and illustration

First minors

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. This number is often denoted M_i,j. The (i, j) cofactor is obtained by multiplying the minor by ${\displaystyle (-1)^{i+j}}$.

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

${\displaystyle {\begin{bmatrix}1&4&7\\3&0&5\\-1&9&11\\\end{bmatrix}}}$

To compute the minor M_2,3 and the cofactor C_2,3, we find the determinant of the above matrix with row 2 and column 3 removed.

${\displaystyle M_{2,3}=\det {\begin{bmatrix}1&4&\Box \\\Box &\Box &\Box \\-1&9&\Box \\\end{bmatrix}}=\det {\begin{bmatrix}1&4\\-1&9\\\end{bmatrix}}=9-(-4)=13}$

So the cofactor of the (2,3) entry is

${\displaystyle \ C_{2,3}=(-1)^{2+3}(M_{2,3})=-13.}$

General definition

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 ${\textstyle {m \choose k}\cdot {n \choose k}}$ 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.^[2][3]

Let ${\displaystyle 1\leq i_{1}<i_{2}<\cdots <i_{k}\leq m}$ and ${\displaystyle 1\leq j_{1}<j_{2}<\cdots <j_{k}\leq n}$ 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. The minor ${\textstyle \det \left((A_{i_{p},j_{q}})_{p,q=1,\ldots ,k}\right)}$ corresponding to these choices of indexes is denoted ${\displaystyle \det _{I,J}A}$ or ${\displaystyle \det A_{I,J}}$ or ${\displaystyle _{I,J}}$ or ${\displaystyle M_{I,J}}$ or ${\displaystyle M_{i_{1},i_{2},\ldots ,i_{k},j_{1},j_{2},\ldots ,j_{k}}}$ or ${\displaystyle M_{(i),(j)}}$ (where the ${\displaystyle (i)}$ denotes the sequence of indexes I, etc.), depending on the source. 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 from the source in question. In this article, we use the inclusive definition of choosing the elements from rows of I and columns of J. The exceptional case is the case of the first minor or the (i, j)-minor described above; in that case, the exclusive meaning ${\textstyle M_{i,j}=\det \left(\left(A_{p,q}\right)_{p\neq i,q\neq j}\right)}$ is standard everywhere in the literature and is used in this article also.

Complement

The complement, B_{ijk...,pqr...}, of a minor, M_{ijk...,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 M_{ijk...,pqr...} have been removed. The complement of the first minor of an element a_ij is merely that element.^[5]

Applications of minors and cofactors

Cofactor expansion of the determinant

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 ${\displaystyle A=(a_{ij})}$, 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. In other words, defining ${\displaystyle C_{ij}=(-1)^{i+j}M_{ij}}$ then the cofactor expansion along the j th column gives:

${\displaystyle det(\mathbf{A})=a_{1j}{C}_{1j}+a_{2j}{C}_{2j}+a_{3j}{C}_{3j}+\cdots <!--\; \cdots \; -->+a_{nj}}$Zdroj:https://en.wikipedia.org?pojem=Minor_(linear_algebra)
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---