Cofactor (linear algebra)

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In linear algebra, the cofactor describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. Specifically the cofactor of an entry of a matrix is the signed minor of that entry.

1 Informal approach to minors and cofactors
2 Definition
3 Example
4 Cofactor expansion
5 Matrix of cofactors
6 Adjugate
7 See also
8 References
9 External links

[edit] Informal approach to minors and cofactors

Finding the minors of a matrix is a multi-step process. First, choose the an entry $a i j$ from the matrix. Then cross out the terms that lie in the equivalent row $i$ and column $j$ . Next, rewrite the matrix without the marked entries. Finally, obtain the determinant of this new matrix. This determinant is termed the minor $M i j$ for entry $a i j$ . Once the minor has been obtained, the cofactor can be determined by multiplying the minor $M i j$ by $( - 1) i + j$ where $i$ and $j$ are the row and column numbers that correspond to the minor $M i j$ .

[edit] Definition

If A is a square matrix, then the minor entry of $a i j$ is denoted by $M i j$ and is defined to be the determinant of the submatrix that remains after the i-th row and the j-th column are deleted from A. The number $( - 1) i + j M i j$ is denoted by $C i j$ and is called the cofactor of $a i j$ .

[edit] Example

Given the matrix

$B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \\ \end{bmatrix}$

suppose we wish to find the cofactor C₂₃. The minor M₂₃ is the determinant of the above matrix with row 2 and column 3 removed.

$M_{23} = \begin{vmatrix} b_{11} & b_{12} & \Box \\ \Box & \Box & \Box \\ b_{31} & b_{32} & \Box \\ \end{vmatrix}$ yields $M_{23} = \begin{vmatrix} b_{11} & b_{12} \\ b_{31} & b_{32} \\ \end{vmatrix} = b_{11}b_{32} - b_{31}b_{12}$

Using the given definition it follows that

$\ C_{23} = (-1)^{2+3}(M_{23})$

$\ C_{23} = (-1)^{5}(b_{11}b_{32} - b_{31}b_{12})$

$\ C_{23} = b_{31}b_{12} - b_{11}b_{32}.$

Note: the vertical lines are an equivalent notation for det(matrix)

[edit] Cofactor expansion

Main article: Laplace expansion

Given the $n\times n$ matrix

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}$

The determinant of A (det(A)) can be written as the sum of its cofactors multiplied by the entries that generated them.

$\ \det(A) = a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} + ... + a_{nj}C_{nj}$

(cofactor expansion along the jth column)

$\ \det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + ... + a_{in}C_{in}$

(cofactor expansion along the ith row)

for any row i and column j in the matrix A.

[edit] Matrix of cofactors

The matrix of cofactors for an n by n matrix A is the matrix that contains the corresponding cofactor Cij in place of every entry a_ij in A

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}$

yields

$\begin{bmatrix} C_{11} & C_{12} & \cdots & C_{1n} \\ C_{21} & C_{22} & \cdots & C_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ C_{n1} & C_{n2} & \cdots & C_{nn} \end{bmatrix}$

where C_ij is the cofactor that corresponds to a_ij.

[edit] Adjugate

Main article: Adjugate matrix

The adjugate matrix is the transpose of the matrix of cofactors and is very useful due to its relation to the inverse of A.

$A^{-1} = \left ( \frac{1}{\det(A)} \right ) \mathrm{adj}(A)$

The matrix of cofactors

$\begin{bmatrix} C_{11} & C_{12} & \cdots & C_{1n} \\ C_{21} & C_{22} & \cdots & C_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ C_{n1} & C_{n2} & \cdots & C_{nn} \end{bmatrix}$

when transposed becomes

$\mathrm{adj}(A) = \begin{bmatrix} C_{11} & C_{21} & \cdots & C_{n1} \\ C_{12} & C_{22} & \cdots & C_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ C_{1n} & C_{2n} & \cdots & C_{nn} \end{bmatrix}.$