代数摁算

\[\begin{vmatrix} A&0\\ -E& B \end{vmatrix} =|A||B|\\ \begin{vmatrix} a_{11}&a_{12}&…&a_{1n}&0&0&…&0\\ a_{21}&a_{22}&…&a_{2n}&0&0&…&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&…&\vdots\\ a_{n1}&a_{n_2}&…&a_{nn}&0&0&…&0\\ -1&0&…&0&b_{11}&b_{12}&…&b_{1n}\\ 0&-1&…&0&b_{21}&b_{22}&…&b_{2n}\\ \vdots&\vdots& &\vdots&\vdots&\vdots& &\vdots\\ 0&0&…&-1&b_{n1}&b_{n2}&…&b_{nn} \end{vmatrix} \]

把第\(n+1\)行的\(a_{11}\)倍加到\(1\)行，第\(n+2\)行的\(a_{12}\)倍加到第\(1\)行……第\(n+n\)行的\(a_{1n}\)倍加到第\(1\)行

\(……\)

\[\begin{vmatrix} 0&0&…&0&\sum_{k=1}^{n}a_{1k}{k1}&\sum_{k=1}^{n}a_{2k}{k2}&…&\sum_{k=1}^{n}a_{nk}{kn}\\ a_{21}&a_{22}&…&a_{2n}&0&0&…&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&…&\vdots\\ a_{n1}&a_{n_2}&…&a_{nn}&0&0&…&0\\ -1&0&…&0&b_{11}&b_{12}&…&b_{1n}\\ 0&-1&…&0&b_{21}&b_{22}&…&b_{2n}\\ \vdots&\vdots& &\vdots&\vdots&\vdots& &\vdots\\ 0&0&…&-1&b_{n1}&b_{n2}&…&b_{nn} \end{vmatrix} \]

再把第\(n+1\)行的\(a_{i,1}\)倍加到第\(i\)行，第\(n+2\)行的第\(a_{i,2}\)倍加到第\(i\)行……第\(n+n\)行的\(a_{i,n}\)倍加到第\(i\)行

其中，\(i=2,3,…,n\)

\[\begin{vmatrix} 0&0&…&0&\sum_{k=1}^{n}a_{1k}b_{k1}&\sum_{k=1}^{n}a_{1k}b_{k2}&…&\sum_{k=1}^{n}a_{1k}b_{kn}\\ 0&0&…&0&\sum_{k=1}^{n}a_{2k}b_{k1}&\sum_{k=1}^{n}a_{2k}b_{k2}&…&\sum_{k=1}^{n}a_{2k}b_{kn}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&…&\vdots\\ 0&0&…&0&\sum_{k=1}^{n}a_{nk}b_{k2}&\sum_{k=1}^{n}a_{nk}b_{k2}&…&\sum_{k=1}^{n}a_{nk}b_{kn}\\ -1&0&…&0&b_{11}&b_{12}&…&b_{1n}\\ 0&-1&…&0&b_{21}&b_{22}&…&b_{2n}\\ \vdots&\vdots& &\vdots&\vdots&\vdots& &\vdots\\ 0&0&…&-1&b_{n1}&b_{n2}&…&b_{nn} \end{vmatrix}\\ = \begin{vmatrix} 0&AB\\ -E&B \end{vmatrix}\\ =|AB| \]

几何胡扯

\(n\)阶段矩阵\(A\)表示对\(n\)阶空间一组线性基\(E\)的线性变换，其中\(E\)的向量组成的\(n\)维超立方体的体积为\(1\)

\(A\)的行列式\(|A|\)表示经过线性变换\(A\)后，基围成的体积的变化情况

如\(|AE|\)表示\(E\)的正交基向量变为\(A\)中的\(n\)行列向量，基向量围成的\(n\)维立方体体积变为\(|A|\)倍

\(|ABE|=1*|B|*|A|=1*|A|*|B|=|BAE|=|BA|\)