第七题行列式求解
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全加到第一列，得
x+y+z x y z
x+y+z 0 z y
x+y+z z 0 x
x+y+z y x 0
同时减去第一行，得
x+y+z x y z
0 -x z-y y-z
0 z-x -y x-z
0 y-x x-y -z
=(x+y+z)×
|-x z-y y-z
z-x -y x-z
y-x x-y -z|
=(x+y+z)×
|-x z-y 0
z-x -y x-y-z
y-x x-y x-y-z|
=(x+y+z)×
|-x z-y 0
z-y -x 0
y-x x-y x-y-z|
=(x+y+z)×(x-y-z)×
|-x z-y
z-y -x|
=(x+y+z)×(x-y-z)×[x²-(z-y)²]
=(x+y+z)×(x-y-z)×(x+z-y)(x-z+y)
x+y+z x y z
x+y+z 0 z y
x+y+z z 0 x
x+y+z y x 0
同时减去第一行，得
x+y+z x y z
0 -x z-y y-z
0 z-x -y x-z
0 y-x x-y -z
=(x+y+z)×
|-x z-y y-z
z-x -y x-z
y-x x-y -z|
=(x+y+z)×
|-x z-y 0
z-x -y x-y-z
y-x x-y x-y-z|
=(x+y+z)×
|-x z-y 0
z-y -x 0
y-x x-y x-y-z|
=(x+y+z)×(x-y-z)×
|-x z-y
z-y -x|
=(x+y+z)×(x-y-z)×[x²-(z-y)²]
=(x+y+z)×(x-y-z)×(x+z-y)(x-z+y)
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