PlanetMath: proof of cofactor expansion

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proof of cofactor expansion

(Proof)

Let $M \in mat_N(K)$ be a $n \times n$ -matrix with entries from a commutative field . Let $e_1, \ldots, e_n$ denote the vectors of the canonical basis of . For the proof we need the following

Lemma: Let $M_{ij}^*$ be the matrix generated by replacing the -th row of by . Then

$\displaystyle \det{M_{ij}^*} =(-1)^{i+j}\det{M_{ij}}$

where $M_{ij}$ is the $(n -1) \times (n-1)$ -matrix obtained from

by removing its

-th row and

-th column.

Proof. By adding appropriate of the

-th row of $M_{ij}^*$ to its remaining rows we obtain a matrix with 1 at position

and 0 at positions

( $k \neq i$ ). Now we apply the permutation

$\displaystyle (1 2) \circ (2 3) \circ\dots \circ ((i -1) i)$

to rows and

$\displaystyle (1 2) \circ (2 3) \circ\dots \circ ((j-1)j)$

to columns of the matrix. The matrix now looks like this:

Row/column 1 is the vector ;
under row 1 and right of column 1 is the matrix $M_{ij}$ .

Since the determinant has changed its sign

times, we have

$\displaystyle \det{M_{ij}^*} =(-1)^{i+j}\det{M_{ij}}.$

Note also that only those permutations $\pi \in S_n$ are for the computation of the determinant of $M_{ij}^*$ where $\pi(i)=j$ . $\qedsymbol$

Now we start out with

$\displaystyle \det{M}$		$\displaystyle =\sum\limits_{\pi \in S_n} \mathrm{sgn} \pi\left(\prod_{j=1}^n m_{j\pi(j)}\right)$
		$\displaystyle =\sum_{k=1}^n m_{ik}\left(\sum\limits_{\pi \in S_n \mid \pi(i)=k}... ...t) \cdot 1 \cdot \left(\prod\limits_{i \le j \le n} m_{j-\pi(j)}\right)\right).$

From the previous lemma, it follows that the associated with $M_{ik}$ is the determinant of $M_{ij}^*$ . So we have

$\displaystyle \det{M} =\sum_{k=1}^n M_{ik}\left((-1)^{i+k}\det{M_{ik}}\right).$

"proof of cofactor expansion" is owned by Thomas Heye.

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Other names:

Laplace expansion

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Cross-references: determinant, right, permutation, column, row, generated by, matrix, canonical basis, vectors, field, commutative

This is version 10 of proof of cofactor expansion, born on 2003-01-16, modified 2005-11-29.
Object id is 3896, canonical name is ProofOfCofactorExpansion.
Accessed 4314 times total.

Classification:

AMS MSC:

15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)

Pending Errata and Addenda

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