cofactor expansion (original) (raw)

We have the following useful formulas for the cofactors of a matrix. First, if we regard det⁡M as a polynomial in the entries Mi⁢j, then we may write

Second, we may regard the determinant of M=(M1,…,Mn) as a multi-linear, skew-symmetric function of its columns:

This point of view leads to the following formula:

where the notation indicates that column j has been replaced by the ith standard vector.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 j. If we regard the determinant as a multi-linear, skew-symmetric function of n row-vectors, then we obtain the analogous cofactor expansion along a row:

Example.

Consider a general 3×3 determinant

| |a1a2a3b1b2b3c1c2c3|=a1⁢b2⁢c3+a2⁢b3⁢c1+a3⁢b1⁢c2-a1⁢b3⁢c2-a3⁢b2⁢c1-a2⁢b1⁢c3. | | -------------------------------------------------------------------------------- |

The above can equally well be expressed as a cofactor expansion along the first row:

| |a1a2a3b1b2b3c1c2c3| | =a1⁢|b2b3c2c3|-a2⁢|b1b3c1c3|+a3⁢|b1b2c1c2| | | --------------------------------------------------------- | --------------------------------------------- | | =a1⁢(b2⁢c3-b3⁢c2)-a2⁢(b1⁢c3-b3⁢c1)+a3⁢(b1⁢c2-b2⁢c1); | |

or along the second column:

| |a1a2a3b1b2b3c1c2c3| | =-a2⁢|b1b3c1c3|+b2⁢|a1a3c1c3|-c2⁢|a1a3b1b3| | | ----------------------------------------------------------- | ----------------------------------------------- | | =-a2⁢(b1⁢c3-b3⁢c1)+b2⁢(a1⁢c3-a3⁢c1)-c2⁢(a1⁢b3-a3⁢b1); | |

or indeed as four other such expansion corresponding to rows 2 and 3, and columns 1 and 3.