(1)

若x=0或y=0, 有兩行相等行列式等於0

當x≠0且y≠0時

D =

r2-r1,r3-r1,r4-r1

1+x 1 1 1

-x -x 0 0

-x 0 y 0

-x 0 0 -y

c1-c2+(x/y)c3-(x/y)c4

x 1 1 1

0 -x 0 0

0 0 y 0

0 0 0 -y

= x(-x)y(-y)= x^2y^2.

(2)

解: D =

c1+5c5, c3-2c5

10 4 1 -1 2

0 0 0 0 1

-13 2 8 3 -4

-3 1 -1 -5 0

17 -3 -6 1 3

按第2行展開 D=(-1)^(2+5)*

10 4 1 -1

-13 2 8 3

-3 1 -1 -5

17 -3 -6 1

c1+10c4, c2+4c4, c3+c4

0 0 0 -1

17 14 11 3

-53 -19 -6 -5

27 1 -5 1

按第1行展開 D = - (-1)^(1+4)*(-1)*

17 14 11

-53 -19 -6

27 1 -5

r1+r2+r3, r3-r2

-9 -4 0

-53 -19 -6

80 20 1

r2+6r3

-9 -4 0

427 101 0

80 20 1

= (-1)*

-9 -4

427 101

= (-1)*(-909 + 4*427)

= -799