cofactor expansion

Let $M$ be an $n\\times n$ matrix with entries $M_{ij}$ that areelements of a commutative ring.Let $m_{ij}$ denote thedeterminant of the $(n-1)\\times(n-1)$ submatrix obtained by deletingrow $i$ 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

C_{ij}=\\frac{\\partial M}{\\partial M_{ij}}

(1)

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:

C_{ij}=\\det(M_{1},\\ldots,\\hat{M_{j}},\\mathbf{e}_{i},\\ldots,M_{n}),

(2)

where the notation indicates that column $j$ has been replaced by the $i$ th standard vector.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 ofcofactors:

	$\\displaystyle\\det(M)$	$\\displaystyle=\\det(M_{1},\\ldots,M_{1j}\\mathbf{e}_{1}+\\cdots+M_{nj}\\mathbf{e}_{%n},\\ldots,M_{n})$
		$\\displaystyle=\\sum_{i=1}^{n}M_{ij}C_{ij},\\quad j=1,\\ldots,n.$

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 theanalogous cofactor expansion along a row:

\\displaystyle\\det(M)

\\displaystyle=\\sum_{i=1}^{n}M_{ji}C_{ji}.

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 alongthe first row:

	$\\displaystyle\\left\|\\begin{matrix}a_{1}&a_{2}&a_{3}\\\\b_{1}&b_{2}&b_{3}\\\\c_{1}&c_{2}&c_{3}\\end{matrix}\\right\|$	$\\displaystyle=a_{1}\\left\|\\begin{matrix}b_{2}&b_{3}\\\\c_{2}&c_{3}\\end{matrix}\\right\|-a_{2}\\left\|\\begin{matrix}b_{1}&b_{3}\\\\c_{1}&c_{3}\\end{matrix}\\right\|+a_{3}\\left\|\\begin{matrix}b_{1}&b_{2}\\\\c_{1}&c_{2}\\end{matrix}\\right\|$
		$\\displaystyle=a_{1}(b_{2}c_{3}-b_{3}c_{2})-a_{2}(b_{1}c_{3}-b_{3}c_{1})+a_{3}(%b_{1}c_{2}-b_{2}c_{1});$

or along the second column:

	$\\displaystyle\\left\|\\begin{matrix}a_{1}&a_{2}&a_{3}\\\\b_{1}&b_{2}&b_{3}\\\\c_{1}&c_{2}&c_{3}\\end{matrix}\\right\|$	$\\displaystyle=-a_{2}\\left\|\\begin{matrix}b_{1}&b_{3}\\\\c_{1}&c_{3}\\end{matrix}\\right\|+b_{2}\\left\|\\begin{matrix}a_{1}&a_{3}\\\\c_{1}&c_{3}\\end{matrix}\\right\|-c_{2}\\left\|\\begin{matrix}a_{1}&a_{3}\\\\b_{1}&b_{3}\\end{matrix}\\right\|$
		$\\displaystyle=-a_{2}(b_{1}c_{3}-b_{3}c_{1})+b_{2}(a_{1}c_{3}-a_{3}c_{1})-c_{2}%(a_{1}b_{3}-a_{3}b_{1});$

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