PlanetMath: cofactor expansion

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cofactor expansion

(Theorem)

Let 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 and column of , and let

$\displaystyle C_{ij} = (-1)^{i+j} m_{ij}.$

The subdeterminants $m_{ij}$ are called the minors of

, 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

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

(1)

Second, we may regard the determinant of $M=(M_1,\ldots, M_n)$ as a multi-linear, skew-symmetric function of its columns:

$\displaystyle \det M = \det(M_1,\ldots, M_n).$

This point of view leads to the following formula:

$\displaystyle C_{ij} = \det(M_1,\ldots, \hat{M_j}, \mathbf{e}_i,\ldots, M_n),$

(2)

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has been replaced by the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:

$\displaystyle \det(M)$	$\displaystyle = \det(M_1,\ldots, M_{1j} \mathbf{e}_1 + \cdots + M_{nj} \mathbf{e}_n , \ldots , M_n)$
	$\displaystyle = \sum_{i=1}^n M_{ij} C_{ij},\quad j=1,\ldots, n.$

The above identity is often called the cofactor expansion of the determinant along column

. 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:

$\displaystyle \det(M)$

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

Example.

Consider a general $3\times 3$ determinant

$\displaystyle \left\vert \begin{matrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \\ ... ...b_2 c_3 + a_2 b_3 c_1 + a_3 b_1 c_2 - a_1 b_3 c_2 - a_3 b_2 c_1 - a_2 b_1 c_3.$

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

$\displaystyle \left\vert\begin{matrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right\vert$	$\displaystyle = a_1 \left\vert \begin{matrix}b_2 & b_3 \\ c_2 & c_3 \end{matrix... ...t + a_3 \left\vert \begin{matrix}b_1 & b_2 \\ c_1 & c_2 \end{matrix}\right\vert$
	$\displaystyle = a_1(b_2 c_3-b_3 c_2) - a_2(b_1 c_3 - b_3 c_1) + a_3(b_1 c_2-b_2c_1);$

or along the second column:

$\displaystyle \left\vert\begin{matrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right\vert$	$\displaystyle = -a_2 \left\vert \begin{matrix}b_1 & b_3 \\ c_1 & c_3 \end{matri... ...t - c_2 \left\vert \begin{matrix}a_1 & a_3 \\ b_1 & b_3 \end{matrix}\right\vert$
	$\displaystyle = -a_2(b_1 c_3-b_3 c_1) + b_2(a_1 c_3 - a_3 c_1) - c_2(a_1 b_3-a_3 b_1);$

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

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"cofactor expansion" is owned by rmilson. [ full author list (3) | owner history (1) ]

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