Using determinant property to calculate (need process): (x, y, x + y; y, x + y, X; X + y, x, y)

Using determinant property to calculate (need process): (x, y, x + y; y, x + y, X; X + y, x, y)

Add the second and third lines to the first line respectively, the first line is 2 * (x + y), extract, get 2 * (x + y) * {1,1,1; y, x + y, X; X + y, x, y};
Then multiply the first line by (- y), (- X-Y) and add it to two or three lines to get 2 * (x + y) * {1,1,1; 0, x, X-Y; 0, - y, - x};
Finally, the original formula = - 2 * (x ^ 3 + y ^ 3)

Calculate the n-stage determinant, please help me answer: calculate the 3-stage determinant | x y x + y | y x + y x | x + y x y|
| x y x+y |
| y x+y x |
|x+y x y |

Direct expansion: = 3YX ^ 2 + 3xy ^ 2-x ^ 3-y ^ 3-3yx ^ 2-3xy ^ 2 = - x ^ 3-y ^ 3

Calculate the following determinant 1 + x 1 + x 1 + Y 1 + Y 1
1+X 1 1 1
1 1+X 1 1
1 1 1+y 1
1 1 1 1+y

If y = 0, determinant 1 and 2 are equal, then determinant = 0
When y ≠ 0
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
2+x+2x/y 1 1 1
0 x 0 0
0 0 y 0
0 0 0 y
Determinant = x * y * y * (2 + X + 2x / y) = x ^ 2Y ^ 2 + 2x ^ 2Y + 2XY ^ 2
It can be seen that when y = 0, the above formula = 0
So determinant = x ^ 2Y ^ 2 + 2x ^ 2Y + 2XY ^ 2