Cofactor (mathematics)

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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 M_i,j formed by deleting the i-th row and j-th column from M (or sometimes just to the submatrix M_i,j itself). The corresponding cofactor is the signed determinant

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

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

${\displaystyle M\cdot \text{adj}M=(\det M)I_n=\text{adj}M\cdot M,}$

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

${\displaystyle M^{-1}=(\det M{)}^{-1}\text{adj}M.}$

Example[edit]

Consider the following example matrix,

${\displaystyle M=\left(\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\\ \end{array}\right).}$

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

${\displaystyle M_{11}=\left|\begin{array}{cc}{b}_2& b_3\\ c_2& c_3\\ \end{array}\right|M_{12}=\left|\begin{array}{cc}{b}_1& b_3\\ c_1& c_3\\ \end{array}\right|M_{13}=\left|\begin{array}{cc}{b}_1& b_2\\ c_1& c_2\\ \end{array}\right|M_{21}=\left|\begin{array}{cc}{a}_2& a_3\\ c_2& c_3\\ \end{array}\right|M_{22}=\left|\begin{array}{cc}{a}_1& a_3\\ c_1& c_3\\ \end{array}\right|}$

${\displaystyle M_{23}=\left|\begin{array}{cc}{a}_1& a_2\\ c_1& c_2\\ \end{array}\right|M_{31}=\left|\begin{array}{cc}{a}_2& a_3\\ b_2& b_3\\ \end{array}\right|M_{32}=\left|\begin{array}{cc}{a}_1& a_3\\ b_1& b_3\\ \end{array}\right|M_{33}=\left|\begin{array}{cc}{a}_1& a_2\\ b_1& b_2\\ \end{array}\right|}$

The adjugate matrix of M is

${\displaystyle \mathrm{a}\mathrm{d}\mathrm{j}M=A=\left(\begin{array}{ccc}{M}_{11}& -M_{21}& M_{31}\\ -M_{12}& M_{22}& -M_{32}\\ M_{13}& -M_{23}& M_{33}\\ \end{array}\right),}$

and the inverse matrix is

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

Indeed,

${\displaystyle \begin{array}{rl}{\left(M{M}^{-1}\right)}_{11}& =|M{|}^{-1}(a_1{M}_{11}-a_2{M}_{12}+a_3{M}_{13})=\frac{|M|}{|M|}=1\\ {\left(M{M}^{-1}\right)}_{21}& =|M{|}^{-1}(b_1{M}_{11}-b_2{M}_{12}+b_3{M}_{13})=|M{|}^{-1}[b_1(b_2{c}_3-b_3{c}_2)-b_2(b_1{c}_3-b_3{c}_1)+b_3(b_1{c}_2-b_2{c}_1)]=0,\\ \end{array}}$

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.