Cofactor matrix

Cofactor matrix

The cofactor matrix of a square matrix M of size n, also called comatrix and noted com(M), is a square matrix of size n defined as follows :

We note c(i, j) the element in row i and column j of the comatrix. Then,

`c(i,j) = (-1)^(i+j) det( M(i,j) )`, where
'det' represents the determinant.
`M(i,j)` is the square submatrix of size n - 1 obtained from M by deleting the row i and the column j. The determinant of M(i,j) is called 'Minor' of matrix M.

c (i, j) is called cofactor of index i, j.

Example of calculating a 2 x 2 comatrix :

`M = [[1,2],[-3,6]]`

We calculate the cofactors as explained above,

`c(1,1) = (-1)^(1+1) det( (6) ) = 6`

`c(1,2) = (-1)^(1+2) det( (-3) ) = 3`

`c(2,1) = (-1)^(2+1) det( (2) ) = -2`

`c(2,2) = (-1)^(2+2) det( (1) ) = 1`

Therefore,

`\text{com(M)} = [[6,3],[-2,1]]`

Example of calculation of a 3x3 comatrix :

`M = [[1,5,4],[-2,7,-2],[9,8,0]]`

By applying the cofactors formula of cofactors, we get,

`c(1,1) = (-1)^(1+1) det([[7,-2],[8,0]]) = 16`

`c(1,2) = (-1)^(1+2) det([[-2,-2],[9,0]]) = -18`

`c(1,3) = (-1)^(1+3) det([[-2,9],[7,8]]) = -79`

`c(2,1) = (-1)^(2+1) det([[5,4],[8,0]]) = 32`

`c(2,2) = (-1)^(2+2) det([[1,4],[9,0]]) = -36`

`c(2,3) = (-1)^(2+3) det([[1,5],[9,8]]) = 37`

`c(3,1) = (-1)^(3+1) det([[5,4],[7,-2]]) = -38`

`c(3,2) = (-1)^(3+2) det([[1,4],[-2,-2]]) = -6`

`c(3,3) = (-1)^(3+3) det([[1,5],[-2,7]]) = 17`

Then we have,

`\text{com(M)} = [[16,-18,-79],[32,-36,37],[-38,-6,17]]`

See also

Matrix determinant
Adjoint Matrix
Linear algebra Calculators