1.2-1.4 行列式的性质(5)

2

|B|

a11 a|A|

r

s

···a1r···a···a1s···a······ a1n a|C|A|,|B|C|

0.

8(

|A|= 21

2r2s2n ···

·················· an1···anr···ans···ann ···a1s···a1r···a1n

|B|= a11 a21

···a2s···a2r···a2n ·················· ···ans

···anr

···

ann ···

an1

···a1r+a1s···a1r+a1s···a1n= a11 a21

···a2r+a2s···a2r+a2s···a2n ···

·················· an1···anr+ans···anr+ans

···

ann ···a1r···a1r+a1s···a 1n

= a11 a21···a2r···a2r+a2s···a2n

···

·················· an1···anr

···anr+ans

···ann ···a1s···a1r+a1s···a 1n

+ a11 a21···a2s···a2r+a2s···a2n

···

·················· an1

···ans···anr+ans···ann ···a 1n

=|A|+

a11···a1r···a1r a21···a2r···a2r

···a2n

···············

······ a

n1···anr···anr···ann ···a 1

n+|B|+

a11···a1s···a1s a21···a2s···a2s

···a2n

···············

······ an1···ans···ans

···ann 0,|C|

r

s

|C|=|A|+|B|=0,

|A|= |B|.r

)

|A|=a1rA1r+a2rA2r+···+anrAnr

5

2

||