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5.矩阵的行列式-相关性质

若存在行列式:

|A|=|a11a12a13...a1na21a22a23...a2na31a32a33...a3n......an1an2an3...ann|

|A|具有以下性质:

5.1 性质1:|A|T=|A|

性质1的证明:

由矩阵转置相关性质可知:

(aij)T=a(ji)

而:

|A|=(1)ta1p1a2p2a3p3...anpn|A|T=(1)tap11ap22ap33...apnn

故:

(1)|A|T=|A|

5.2 性质2:将行列式任意两行(或两列)进行互换,行列式变号

性质2的证明:

|A|=(1)ta1p1a2p2a3p3...ajpj...akpk...anpn

设将|A|中第j行和第k行进行互换形成的行列式为|Akj|

相对|A|而言,|Akj|中的全排列变为:p1p2p3...pk...pj...pn,即产生(或减少)了1个逆序数:

|Akj|=(1)t±1a1p1a2p2a3p3...akpk...ajpj...anpn

(2)|Akj|=|A|

5.3 性质3:若行列式任意两行的值完全相同,则行列式的值为0

性质3的证明:

根据性质2可知:|Akj|=|A|

|A|k,j两行的值完全相同,则:

|Akj|=|A||A|=|A|

(3)|A|=0

5.4 性质4:λ|A|=λ 乘以 |A|中任意一行(或任意一列)的元素

性质4的证明:

|A|=(1)ta1p1a2p2a3p3...anpn可知:

λ|A|=(1)tλa1p1a2p2a3p3...anpn

由性质1可知:|A|=|A|T

λ|A|=λ|A|T=(1)tap11ap22ap33...apnn

又知p1p2p3......pn 是|A|中的全排列,故aipi表示第i行任一元素,apii表示第i列任一元素(i=1,2,3...,n),则:

λ|A|=(1)t(λa1p1)a2p2a3p3...anpn=(1)ta1p1a2p2a3p3...(λanpn)

=λ|A|T=(1)t(λap11)ap22ap33...apnn......=λ|A|T=(1)tap11ap22ap33...(λapnn)......

λ|A|=|λa11λa12λa13...λa1na21a22a23...a2na31a32a33...a3n......an1an2an3...ann|

(4)=|a11a12a13...a1na21a22a23...a2na31a32a33...a3n......λan1λan2λan3...λann|

=λ|A|T=|λa11a12a13...a1nλa21a22a23...a2nλa31a32a33...a3n......λan1an2an3...ann|

......

(5)=|a11a12a13...λa1na21a22a23...λa2na31a32a33...λa3n......an1an2an3...λann|

5.5 性质5:若行列式任意两行(或两列)元素成比例,则行列值为0

|A|存在第x行:

|A|=|a11a12a13...a1na21a22a23...a2na31a32a33...a3n......ax1ax2ax3...axn......an1an2an3...ann|

|A|中满足:a1i=λaxi(i=1,2,3,...,n),则|A|可写为如下形式:

|A|=|λax1λax2λax3...λaxna21a22a23...a2na31a32a33...a3n......ax1ax2ax3...axn......an1an2an3...ann|

=λ|ax1ax2ax3...axna21a22a23...a2na31a32a33...a3n......ax1ax2ax3...axn......an1an2an3...ann|

则根据性质4,可得:

λ|ax1ax2ax3...axna21a22a23...a2na31a32a33...a3n......ax1ax2ax3...axn......an1an2an3...ann|=λ0=0(6)

5.6 性质6:行列式的两行(或两列)相加

|A|存在第z行:

|A|=|a11a12a13...a1na21a22a23...a2na31a32a33...a3n......az1az2az3...azn......an1an2an3...ann|

若第z行元素均满足:azi=axi+ayi(i=1,2,3,...,n)
则:

|A|=(1)ta1p1a2p2...azpz...anpn=(1)ta1p1a2p2...(axpx+aypy)...anpn=(1)ta1p1a2p2...axpx...anpn+(1)ta1p1a2p2...aypy...anpn

|A|可具有以下性质:

(7)|A|=|a11a12a13...a1na21a22a23...a2na31a32a33...a3n......az1az2az3...azn......an1an2an3...ann|=|a11a12a13...a1na21a22a23...a2na31a32a33...a3n......ax1ax2ax3...axn......an1an2an3...ann|+|a11a12a13...a1na21a22a23...a2na31a32a33...a3n......ay1ay2ay3...ayn......an1an2an3...ann|

同理,可通过性质1证得以下性质(过程略):

|a11a12a13...a1z...a1na21a22a23...a2z...a2na31a32a33...a3z...a3n......an1an2an3...anz...ann|=|a11a12a13...a1x...a1na21a22a23...a2x...a2na31a32a33...a3x...a3n......an1an2an3...anx...ann|+|a11a12a13...a1y...a1na21a22a23...a2y...a2na31a32a33...a3y...a3n......an1an2an3...any...ann|

5.6.1 性质6推论:使行列式中两行(或两列)相加,但保持行列式值不变的方法:

设行列式|A|中存在第x列、第y列:

|A|=|a11a12a13...a1x...a1y...a1na21a22a23...a2x...a2y...a2na31a32a33...a3x...a3y...a3n......an1an2an3...anx...any...ann|

若使第x列的数值产生以下变化,则|A|的值保持不变:

(8)aix(aix+λaiy)(i=1,2,3,...,n)

则新生成的行列式|A|为:

|A|=|a11a12a13...(a1x+λa1y)...a1y...a1na21a22a23...(a2x+λa2y)...a2y...a2na31a32a33...(a3x+λa3y)...a3y...a3n......an1an2an3...(anx+λany)...any...ann|

根据性质6可得:

|A|=|a11a12a13...a1x...a1y...a1na21a22a23...a2x...a2y...a2na31a32a33...a3x...a3y...a3n......an1an2an3...anx...any...ann|+|a11a12a13...λa1y...a1y...a1na21a22a23...λa2y...a2y...a2na31a32a33...λa3y...a3y...a3n......an1an2an3...λany...any...ann|

|A|=|A|+0=|A|

同理,设行列式|A|中存在第x行、第y行,若使x行产生以下变化,则|A|的值保持不变:

(9)axj(axi+λayi)(i=1,2,3,...,n)

5.7 性质7:|A||B|=|AB|

(证明过程参考后续知识点)

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