行列式的向量形式
行列式公式
[|A| = egin{vmatrix}
a_{11} & a_{12} & cdots & a_{1n}\
a_{21} & a_{22} & cdots & a_{2n}\
vdots & vdots & ddots & vdots\
a_{n1} & a_{n2} & cdots & a_{nn}\
end{vmatrix}]
行向量表示
令(alpha_i = (a_{i1} , a_{i2} , cdots , a_{in}), iin(1,2,cdots,n)), 则行列式可以表示为
[|A| = egin{vmatrix}
a_{11} & a_{12} & cdots & a_{1n}\
a_{21} & a_{22} & cdots & a_{2n}\
vdots & vdots & ddots & vdots\
a_{n1} & a_{n2} & cdots & a_{nn}\
end{vmatrix} =
egin{vmatrix} alpha_1 \ alpha_2 \ vdots \ alpha_n end{vmatrix}]
性质2(0向量)
[|A|=
egin{vmatrix} alpha_1 \ vdots \ 0 \ vdots \ alpha_n end{vmatrix} =0]
性质3(某一向量的倍数)
[|B|=
egin{vmatrix} alpha_1 \ vdots \ calpha_i \ vdots \ alpha_n end{vmatrix} =cegin{vmatrix} alpha_1 \ vdots \ alpha_i \ vdots \ alpha_n end{vmatrix} = c|A|]
性质4(两行互换)
[|B|=
egin{vmatrix} alpha_1 \ vdots \ alpha_j \ vdots \ alpha_i \ vdots \ alpha_n end{vmatrix} = -egin{vmatrix} alpha_1 \ vdots \ alpha_i \ vdots \ alpha_j \ vdots \ alpha_n end{vmatrix} = -|A|]
性质5(向量加法c=a+b)
[|C|=
egin{vmatrix} alpha_1 \ vdots \ a+b \ vdots \ alpha_n end{vmatrix} =egin{vmatrix} alpha_1 \ vdots \ a \ vdots \ alpha_n end{vmatrix}
+egin{vmatrix} alpha_1 \ vdots \ b \ vdots \ alpha_n end{vmatrix}
=|A|+|B|]
性质6(两行成比例)
[|B|=
egin{vmatrix} alpha_1 \ vdots \ a \ vdots \ ca \ vdots \ alpha_n end{vmatrix} = cegin{vmatrix} alpha_1 \ vdots \ a \ vdots \ a \ vdots \ alpha_n end{vmatrix} = 0]
性质7(倍加)
[|B|=
egin{vmatrix} alpha_1 \ vdots \ a \ vdots \ ca+b \ vdots \ alpha_n end{vmatrix} = egin{vmatrix} alpha_1 \ vdots \ a \ vdots \ ca \ vdots \ alpha_n end{vmatrix} + egin{vmatrix} alpha_1 \ vdots \ a \ vdots \ b \ vdots \ alpha_n end{vmatrix} = |A|]
列向量表示
令(eta_i = egin{pmatrix}a_{1i} \ a_{2i} \ vdots \ a_{ni}end{pmatrix}, iin(1,2,cdots,n)), 则行列式可以表示为
[|A| = egin{vmatrix}
a_{11} & a_{12} & cdots & a_{1n}\
a_{21} & a_{22} & cdots & a_{2n}\
vdots & vdots & ddots & vdots\
a_{n1} & a_{n2} & cdots & a_{nn}\
end{vmatrix} =
egin{vmatrix} eta_1 & eta_2 & cdots & eta_n end{vmatrix}]
性质2(0向量)
[|A|=
egin{vmatrix} eta_1 & cdots & 0 & cdots & eta_n end{vmatrix} =0]
性质3(某一向量的倍数)
[|B|=
egin{vmatrix} eta_1 & cdots & ceta_i & cdots & eta_n end{vmatrix} ]
[=cegin{vmatrix} eta_1 & cdots & eta_i & cdots & eta_n end{vmatrix}
]
[= c|A|
]
性质4(两行互换)
[|B|=
egin{vmatrix} eta_1 & cdots & eta_j & cdots & eta_i & cdots & eta_n end{vmatrix}]
[= -egin{vmatrix} eta_1 & cdots & eta_i & cdots & eta_j & cdots & eta_n end{vmatrix}
]
[= -|A|
]
性质5(向量加法c=a+b)
[|C|=
egin{vmatrix} eta_1 & cdots & a+b & cdots & eta_n end{vmatrix} ]
[=egin{vmatrix} eta_1 & cdots & a & cdots & eta_n end{vmatrix}
+egin{vmatrix} eta_1 & cdots & b & cdots & eta_n end{vmatrix}
]
[=|A|+|B|
]
性质6(两行成比例)
[|B|=
egin{vmatrix} eta_1 & cdots & a & cdots & ca & cdots & eta_n end{vmatrix} ]
[= cegin{vmatrix} eta_1 & cdots & a & cdots & a & cdots & eta_n end{vmatrix}
]
[= 0
]
性质7(倍加)
[|B|=
egin{vmatrix} eta_1 & cdots & a & cdots & ca+b & cdots & eta_n end{vmatrix} ]
[= egin{vmatrix} eta_1 & cdots & a & cdots & ca & cdots & eta_n end{vmatrix} + egin{vmatrix} eta_1 & cdots & a & cdots & b & cdots & eta_n end{vmatrix}
]
[= |A|
]