proof of cofactor expansion
Let $M\in ma{t}_{N}(K)$ be a $n\times n$matrix with entries from a commutative^{} field $K$. Let ${e}_{1},\mathrm{\dots},{e}_{n}$ denote the vectors of the canonical basis of ${K}^{n}$. For the proof we need the following
Lemma: Let ${M}_{ij}^{*}$ be the matrix generated by replacing the $i$th row of $M$ by ${e}_{j}$. Then
$$det{M}_{ij}^{*}={(1)}^{i+j}det{M}_{ij}$$ 
where ${M}_{ij}$ is the $(n1)\times (n1)$matrix obtained from $M$ by removing its $i$th row and $j$th column.
Proof.
By adding appropriate of the $i$th row of ${M}_{ij}^{*}$ to its remaining rows we obtain a matrix with 1 at position $(i,j)$ and 0 at positions $(k,j)$ ($k\ne i$). Now we apply the permutation^{}
$$(12)\circ (23)\circ \mathrm{\dots}\circ ((i1)i)$$ 
to rows and
$$(12)\circ (23)\circ \mathrm{\dots}\circ ((j1)j)$$ 
to columns of the matrix. The matrix now looks like this:

•
Row/column 1 is the vector ${e}_{1}$;

•
under row 1 and right of column 1 is the matrix ${M}_{ij}$.
Since the determinant^{} has changed its sign $i+j2$ times, we have
$$det{M}_{ij}^{*}={(1)}^{i+j}det{M}_{ij}.$$ 
Note also that only those permutations $\pi \in {S}_{n}$ are for the computation of the determinant of ${M}_{ij}^{*}$ where $\pi (i)=j$. ∎
Now we start out with
$detM$  $={\displaystyle \sum _{\pi \in {S}_{n}}}\mathrm{sgn}\pi \left({\displaystyle \prod _{j=1}^{n}}{m}_{j\pi (j)}\right)$  
$={\displaystyle \sum _{k=1}^{n}}{m}_{ik}\left({\displaystyle \sum _{\pi \in {S}_{n}\mid \pi (i)=k}}\mathrm{sgn}\pi \left({\displaystyle \prod _{1\le j\le i}}{m}_{j\pi (j)}\right)\cdot 1\cdot \left({\displaystyle \prod _{i\le j\le n}}{m}_{j\pi (j)}\right)\right).$ 
From the previous lemma, it follows that the associated with ${M}_{ik}$ is the determinant of ${M}_{ij}^{*}$. So we have
$$detM=\sum _{k=1}^{n}{M}_{ik}\left({(1)}^{i+k}det{M}_{ik}\right).$$ 
Title  proof of cofactor expansion 

Canonical name  ProofOfCofactorExpansion 
Date of creation  20130322 13:22:08 
Last modified on  20130322 13:22:08 
Owner  Thomas Heye (1234) 
Last modified by  Thomas Heye (1234) 
Numerical id  13 
Author  Thomas Heye (1234) 
Entry type  Proof 
Classification  msc 15A15 
Synonym  Laplace expansion^{} 