Question

一道简单的行列式计算题

|x y x+y|
|y x+y y|
|x+y y x|这个行列式怎么解？不是要答案0，麻烦说一下过程~

Answer 1

按照3阶行列式的对角线法直接计算化简即得。
答案是0吗?

Answer 2

|x y x+y|
|y x+y y|
|x+y y x|
=x|x+y y|+y(-1)| y y|+(x+y)| y x+y|
| y x| |x+y x| |x+y y |
=x(x²+xy-y²)-y(xy-xy-y²)+(x+y)(y²-x²-2xy-y²)
=x(x²+xy-y²)-y(-y²)+(x+y)(-x²-2xy)
=x³+x²y-xy²+y³-x³-x²y-2x²y-2xy²
=y³-2x²y-3xy²
=y(y²-2x²-3xy)