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矩阵乘积的行列式

矩阵乘积的行列式. 几何观点 矩阵 A 决定 线性变换 f: X AX 所有图形的 n 维体积变为原来的 detA 倍 . g:Y BY. n 维体积  detB 倍 . gf: X(BA)X, n 维体积  原来的 det(BA) 倍 = (detB)(detA) 倍. 代数证明. B AB, det(AB)= a detB 情况 1. A 是初等矩阵 : A: 互换两行 , a= -1=det A A: 某行乘 l 倍 , a= l =det A A: 某行的 l 倍加到另一行 , a=1=detA

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矩阵乘积的行列式

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  1. 矩阵乘积的行列式 几何观点 矩阵A决定线性变换f: XAX 所有图形的n维体积变为原来的 detA倍. g:YBY. n维体积 detB倍. gf: X(BA)X, n维体积 原来的det(BA)倍 = (detB)(detA)倍.

  2. 代数证明 BAB, det(AB)= a detB 情况1. A是初等矩阵: A:互换两行, a= -1=det A A:某行乘l倍, a=l=det A A:某行的l倍加到另一行, a=1=detA 情况2. A不可逆: detA=0, AB不可逆, det(AB)=0=(detA)(detB).

  3. 代数证明 情况3.A可逆. A= Ps…P2P1 , 其中 Ps , …, P2, P1是初等矩阵 det(AB)=det Ps…detP2 detP1 detB (1) 取 B=I 得 detA = det Ps…detP2 detP1 代入(1) 得 det(AB) = (detA)(detB)

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