Cofactor
 

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.
The number
http://upload.wikimedia.org/math/2/4...94aea7ed47.png is denoted by Cij and is 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
http://upload.wikimedia.org/math/7/f...eae9b6952c.png 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.
http://upload.wikimedia.org/math/9/3...9ef9b99f70.png yields http://upload.wikimedia.org/math/b/c...73c208d550.png Using the given definition it follows that
http://upload.wikimedia.org/math/b/0...514f7a3de7.png http://upload.wikimedia.org/math/4/6...ea148bad9e.png http://upload.wikimedia.org/math/6/3...1c6ac1de2e.png Note: the vertical lines are an equivalent notation for det(matrix)

Matrix of cofactors


The matrix of cofactors for an http://upload.wikimedia.org/math/6/0...149a11e90b.png matrix A is the matrix whose (i,j) entry is the cofactor Cij of A. For instance, if A is
http://upload.wikimedia.org/math/f/f...479bcf7f14.png the cofactor matrix of A is
http://upload.wikimedia.org/math/5/0...d776a4e785.png where Cij is the cofactor of aij.