Cofactor (linear algebra)

In linear algebra, the cofactor (sometimes called adjunct, see below) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. Specifically the cofactor of the (i, j) entry of a matrix, also known as the (i, j) cofactor of that matrix, is the signed minor of that entry.

Contents

Informal approach to minors and cofactors

Finding the minors of a matrix A is a multi-step process:

  1. Choose an entry aij from the matrix.
  2. Cross out the entries that lie in the corresponding row i and column j.
  3. Rewrite the matrix without the marked entries.
  4. Obtain the determinant Mij of this new matrix.

Mij is termed the minor for entry aij.

If i + j is an even number, the cofactor Cij of aij coincides with its minor:

C_{ij} = M_{ij}. \,

Otherwise, it is equal to the additive inverse of its minor:

C_{ij} = -M_{ij}. \,

Formal definition

If A is a square matrix, then the minor of its entry aij, also known as the i,j, or (i,j), or (i,j)th minor of A, is denoted by Mij and is defined to be the determinant of the submatrix obtained by removing from A its i-th row and j-th column.

It follows:

C_{ij}=(-1)^{i+j} M_{ij} \,

and Cij called the cofactor of aij, also referred to as the i,j, (i,j) or (i,j)th cofactor of A.

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)

Cofactor 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 (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.

Cofactor expansion along the jth column:

\ \det(A) = a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} + ... + a_{nj}C_{nj} = \sum_{i=1}^{n} a_{ij} C_{ij}

Cofactor expansion along the ith row:

\ \det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + ... + a_{in}C_{in} = \sum_{j=1}^{n} a_{ij} C_{ij}

Matrix of cofactors

The matrix of cofactors for an n\times n matrix A is the matrix whose (i,j) entry is the cofactor Cij of A. For instance, if A is

 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 cofactor matrix of A is

 C = \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 Cij is the cofactor of aij.

Adjugate

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


\mathbf{A}^{-1} = \frac{1}{\det \mathbf{A}} \mbox{adj}(\mathbf{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}.

A remark about different notations

In some books, including the so called "bible of matrix theory"[1] instead of cofactor the term adjunct is used. Moreover, it is denoted as Aij and defined in the same way as cofactor:

\mathbf{A}_{ij} = (-1)^{i+j} \mathbf{M}_{ij}

Using this notation the inverse matrix is written this way:

\mathbf{A}^{-1} = \frac{1}{\det(A)}\begin{bmatrix}
    A_{11}  & A_{21} & \cdots &   A_{n1}   \\
    A_{12}  & A_{22} & \cdots &   A_{n2}   \\
  \vdots & \vdots & \ddots & \vdots \\ 
    A_{1n}  & A_{2n} & \cdots &  A_{nn}
\end{bmatrix}

Keep in mind that adjunct is not adjugate or adjoint.

See also

References

  1. ^ Felix Gantmacher, Theory of matrices (1st ed., original language is Russian), Moscow: State Publishing House of technical and theoretical literature, 1953, p.491,
  • Anton, Howard; Rorres, Chris (2005), Elementary Linear Algebra (9th ed.), John Wiley and Sons, ISBN 0-471-66959-8 

External links


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