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

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where the notation indicates that column $j$ has been replaced by the $i$ th standard vector.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

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

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 the analogous 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 along the 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.

Title	cofactor expansion
Canonical name	CofactorExpansion
Date of creation	2013-03-22 12:01:07
Last modified on	2013-03-22 12:01:07
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	17
Author	rmilson (146)
Entry type	Theorem
Classification	msc 15A15
Synonym	Laplace expansion
Synonym	cofactor
Synonym	minor
Synonym	subdeterminant
Related topic	SarrusRule