设\(\mathbfU\)是\(n\timesm\)阶矩阵，\(\mathbfV\)是\(m\timesn\)阶矩阵，\(\mathbfI_x\)是\(x\)阶单位矩阵。那么矩阵行列式引理的一个核心等式是：\(|\mathbfI_n+\mathbfU\mathbfV|=|\mathbfI_m+\mathbfV\mathbfU|\)。证明考虑分块矩阵乘法，有等式：\[\begin{p

#https://www.chilimath.com/lessons/advanced-algebra/cramers-rule-with-two-variables#https://en.wikipedia.org/wiki/Cramer%27s_rulefromtypingimportList,Tupledefcramers_rule_2x2(equation1:List[int],equation2:List[int])->Tuple[float,float]:

由Amitsur-Levitzki定理，当\(n\ge2k\)时，答案为\(0\)矩阵。否则我们考虑答案矩阵的某一位\(b_{i,j}\)，其必然由某些路径\(i=p_0\top_1\to\\cdots\top_n=j\)贡献而来，一条路径的贡献为\(\text{sgn}(\sigma)\cdot\prod\limits_{i=1}^nA_{\sigma(i),p_{i-1},p_{i}}\)。

//createdon23.12.11目录PfaffianandDeterminantof\(\sumA_iz^i\)PfaffianandDeterminantof\(\sumA_iz^i\)求斜对称矩阵的Pfaffian即\(\mathrm{Pf}(A)\)，主要用到了性质\(\mathrm{Pf}(A)|B|=\mathrm{Pf}(BAB^{T})\)。我们对矩阵进行消元。若第一列全\(0\)，则