线性代数的计算中往往乘法不满足交换律
满足交换律的运算-行列式(determinant)与迹(trace)
t r ( A T ) = t r ( A ) t r ( A + B ) = t r ( A ) + t r ( B ) t r ( A B ) = t r ( B A ) ( A B 为 n × n , B A 为 m × m ) tr(A^T)=tr(A)\ tr(A+B)=tr(A)+tr(B)\ tr(AB)=tr(BA)(AB为n×n,BA为m×m) tr(AT)=tr(A)tr(A+B)=tr(A)+tr(B)tr(AB)=tr(BA)(AB为n×n,BA为m×m)
d e t ( A B ) = d e t ( A ) ∗ d e t ( B ) = d e t ( B A ) det(AB)=det(A)*det(B)=det(BA) det(AB)=det(A)∗det(B)=det(BA)
行列式乘法可交换的初等变换证明：
( E B 0 E ) 为 列 初 等 便 换 矩 阵 ( A A B − E 0 ) = ( A O − E B ) ( E B 0 E ) ∣ A O − E B ∣ = ∣ A ∣ ∣ B ∣ ∣ A A B − E O ∣ = ( − 1 ) n 2 ∣ − E ∣ ∣ A B ∣ = ∣ A B ∣ left( begin{array} { l l } { E } & { B } \ { 0} & { E} end{array}right)为列初等便换矩阵\ left( begin{array} { l l } { A } & { AB } \ { -E } & { 0} end{array}right)= left( begin{array} { l l } { A } & { O } \ { -E } & {B} end{array}right) left( begin{array} { l l } { E } & { B } \ { 0} & { E} end{array}right)\ left| begin{array} { l l } { A } & { O } \ { -E } & { B} end{array}right|=|A||B|\ left| begin{array} { l l } { A } & {AB}\{ -E } & { O } \ end{array}right|=(-1)^{n^2}|-E||AB|=|AB|\ (E0BE)为列初等便换矩阵(A−EAB0)=(A−EOB)(E0BE)∣∣∣∣A−EOB∣∣∣∣=∣A∣∣B∣∣∣∣∣A−EABO∣∣∣∣=(−1)n2∣−E∣∣AB∣=∣AB∣
矩阵的初等变换与"打洞"技巧