One method for computing the determinant is called cofactor expansion.
To define the determinant in the framework of cofactors, one proceeds with an inductive or recursive definition. In such a definition, we give an explicit formula in the case ; then prior to defining the determinant for matirices, we assume that the determinant has already been given for matrices. Note that if we are given an matrix , all of its minors will be of dimensions , for which we can assume the determinant has already been defined. For indices , define the cofactor of to be Then
This is sometimes also referred to as cofactor expansion along the first row.
More generally, cofactor expansion can be easily applied to an arbitrary matrix to recover the usual expression for the determinant in that case.
The following gives an example of how one would use the definition above to compute the determinant of a matrix.