What is the cofactor of a matrix?

The cofactor of a matrix is a scalar value obtained by multiplying a minor by a corresponding sign.

In linear algebra, a cofactor is a scalar value that is associated with each element of a square matrix. To find the cofactor of an element, we first need to find the minor of that element. The minor of an element is the determinant of the submatrix obtained by deleting the row and column containing that element. For example, the minor of the element a11 in a 3x3 matrix A is given by:

minor(a11) = |a22 a23| = a22a33 - a23a32
|a32 a33|

Once we have found the minor of an element, we multiply it by a corresponding sign to obtain the cofactor. The sign of the cofactor is given by (-1)^(i+j), where i and j are the row and column indices of the element. For example, the cofactor of the element a11 in a 3x3 matrix A is given by:

cofactor(a11) = (-1)^(1+1) minor(a11) = a22a33 - a23a32

Cofactors are useful in finding the inverse of a matrix. The inverse of a matrix A is given by:

A^-1 = (1/|A|) adj(A)

where |A| is the determinant of A and adj(A) is the adjugate of A. The adjugate of A is obtained by taking the transpose of the matrix of cofactors of A. For example, the adjugate of a 3x3 matrix A is given by:

adj(A) = |cofactor(a11) cofactor(a21) cofactor(a31)|
|cofactor(a12) cofactor(a22) cofactor(a32)|
|cofactor(a13) cofactor(a23) cofactor(a33)|

In summary, the cofactor of a matrix is a scalar value obtained by multiplying a minor by a corresponding sign. Cofactors are useful in finding the inverse of a matrix.