The subdeterminants are called the minors of , and the are called the cofactors.
We have the following useful formulas for the cofactors of a matrix. First, if we regard as a polynomial in the entries , then we may write
This point of view leads to the following formula:
where the notation indicates that column has been replaced by the th standard vector.
As a consequence, we obtain the following representation of the determinant in terms of cofactors:
The above identity is often called the cofactor expansion of the determinant along column . If we regard the determinant as a multi-linear, skew-symmetric function of row-vectors, then we obtain the analogous cofactor expansion along a row:
Consider a general determinant
The above can equally well be expressed as a cofactor expansion along the first row:
or along the second column:
or indeed as four other such expansion corresponding to rows 2 and 3, and columns 1 and 3.
|Date of creation||2013-03-22 12:01:07|
|Last modified on||2013-03-22 12:01:07|
|Last modified by||rmilson (146)|