cofactor expansion

Let $M$ be an $n\times n$ matrix with entries $M_{ij}$ that are elements of a commutative ring. Let $m_{ij}$ denote the determinant of the $(n-1)\times(n-1)$ submatrix obtained by deleting row $i$ and column $j$ of $M$ , and let $$C_{ij} = (-1)^{i+j} m_{ij}.$$ The subdeterminants $m_{ij}$ are called the minors of $M$ , and the $C_{ij}$ are called the cofactors.

We have the following useful formulas for the cofactors of a matrix. First, if we regard $\det M$ as a polynomial in the entries $M_{ij}$ , then we may write \begin{equation} C_{ij} = \frac{\partial M}{\partial M_{ij}} \end{equation}Second, we may regard the determinant of $M=(M_1,\ldots, M_n)$ as a multi-linear, skew-symmetric function of its columns:$$ \det M = \det(M_1,\ldots, M_n)$$ This point of view leads to the following formula: \begin{equation} C_{ij} = \det(M_1,\ldots, \hat{M_j}, \be_i,\ldots, M_n), \end{equation}where the notation indicates that column $j$ has been replaced by the $i$ th standard vector.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 $j$ . 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 $3\times 3$ determinant $$ \left| \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right|= a_1 b_2 c_3 + a_2 b_3 c_1 + a_3 b_1 c_2 - a_1 b_3 c_2 - a_3 b_2 c_1 - a_2 b_1 c_3. $$ 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.