Cofactor (mathematics)

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This editable Main Article is under development and subject to a disclaimer.

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In mathematics, a cofactor is a component of a matrix computation of the matrix determinant.

Let M be a square matrix of size n. The (i,j) minor refers to the determinant of the (n-1)×(n-1) submatrix Mi,j formed by deleting the i-th row and j-th column from M (or sometimes just to the submatrix Mi,j itself). The corresponding cofactor is the signed determinant

( − 1 ) i + j det M i , j . {\displaystyle (-1)^{i+j}\det M_{i,j}.\,}

The adjugate matrix adj M is the square matrix whose (i,j) entry is the (j,i) cofactor. We have

M ⋅ adj ⁡ M = ( det M ) I n = adj ⁡ M ⋅ M , {\displaystyle M\cdot \mathop {\mbox{adj}} M=(\det M)I_{n}=\mathop {\mbox{adj}} M\cdot M,\,}

which encodes the rule for expansion of the determinant of M by any the cofactors of any row or column. This expression shows that if det M is invertible, then M is invertible and the matrix inverse is determined as

M − 1 = ( det M ) − 1 adj ⁡ M . {\displaystyle M^{-1}=(\det M)^{-1}\mathop {\mbox{adj}} M.\,}

## Example[edit]

Consider the following example matrix,

M = ( a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ) . {\displaystyle M={\begin{pmatrix}a_{1}&a_{2}&a_{3}\\\b_{1}&b_{2}&b_{3}\\\c_{1}&c_{2}&c_{3}\\\\\end{pmatrix}}.}

Its minors are the determinants (bars indicate a determinant):

M 11 = | b 2 b 3 c 2 c 3 | M 12 = | b 1 b 3 c 1 c 3 | M 13 = | b 1 b 2 c 1 c 2 | M 21 = | a 2 a 3 c 2 c 3 | M 22 = | a 1 a 3 c 1 c 3 | {\displaystyle M_{11}={\begin{vmatrix}b_{2}&b_{3}\\\c_{2}&c_{3}\\\\\end{vmatrix}}\quad M_{12}={\begin{vmatrix}b_{1}&b_{3}\\\c_{1}&c_{3}\\\\\end{vmatrix}}\quad M_{13}={\begin{vmatrix}b_{1}&b_{2}\\\c_{1}&c_{2}\\\\\end{vmatrix}}\quad M_{21}={\begin{vmatrix}a_{2}&a_{3}\\\c_{2}&c_{3}\\\\\end{vmatrix}}\quad M_{22}={\begin{vmatrix}a_{1}&a_{3}\\\c_{1}&c_{3}\\\\\end{vmatrix}}\quad }

M 23 = | a 1 a 2 c 1 c 2 | M 31 = | a 2 a 3 b 2 b 3 | M 32 = | a 1 a 3 b 1 b 3 | M 33 = | a 1 a 2 b 1 b 2 | {\displaystyle M_{23}={\begin{vmatrix}a_{1}&a_{2}\\\c_{1}&c_{2}\\\\\end{vmatrix}}\quad M_{31}={\begin{vmatrix}a_{2}&a_{3}\\\b_{2}&b_{3}\\\\\end{vmatrix}}\quad M_{32}={\begin{vmatrix}a_{1}&a_{3}\\\b_{1}&b_{3}\\\\\end{vmatrix}}\quad M_{33}={\begin{vmatrix}a_{1}&a_{2}\\\b_{1}&b_{2}\\\\\end{vmatrix}}\quad }

The adjugate matrix of M is

a d j M = A = ( M 11 − M 21 M 31 − M 12 M 22 − M 32 M 13 − M 23 M 33 ) , {\displaystyle \mathrm {adj} M=A={\begin{pmatrix}M_{11}&-M_{21}&M_{31}\\\\-M_{12}&M_{22}&-M_{32}\\\M_{13}&-M_{23}&M_{33}\\\\\end{pmatrix}},}

and the inverse matrix is

M − 1 = | M | − 1 A . {\displaystyle M^{-1}=|M|^{-1}A\,.}

Indeed,

( M M − 1 ) 11 = | M | − 1 ( a 1 M 11 − a 2 M 12 + a 3 M 13 ) = | M | | M | = 1 ( M M − 1 ) 21 = | M | − 1 ( b 1 M 11 − b 2 M 12 + b 3 M 13 ) = | M | − 1 [ b 1 ( b 2 c 3 − b 3 c 2 ) − b 2 ( b 1 c 3 − b 3 c 1 ) + b 3 ( b 1 c 2 − b 2 c 1 ) ] = 0 , {\displaystyle {\begin{aligned}\left(M\;M^{-1}\right)_{11}&=|M|^{-1}\left(a_{1}M_{11}-a_{2}M_{12}+a_{3}M_{13}\right)={\frac {|M|}{|M|}}=1\\\\\left(M\;M^{-1}\right)_{21}&=|M|^{-1}\left(b_{1}M_{11}-b_{2}M_{12}+b_{3}M_{13}\right)=|M|^{-1}\left[b_{1}(b_{2}c_{3}-b_{3}c_{2})-b_{2}(b_{1}c_{3}-b_{3}c_{1})+b_{3}(b_{1}c_{2}-b_{2}c_{1})\right]=0,\\\\\end{aligned}}}

and the other matrix elements of the product follow likewise.

## References[edit]

* C.W. Norman (1986). Undergraduate Algebra: A first course. Oxford University Press, 306,310,315. ISBN 0-19-853248-2.