Calculate determinant: A ^ n (A-1) ^ n (a-n)^n a Calculate the determinant: a^n (a-1)^n …… …… (a-n)^n a^n-1 (a-1)^n-1 …… …… (a-n)^n-1 …… …… …… …… …… a^n (a-1)^n …… …… (a-n)^n a (a-1) …… …… (a-n) 1 1 …… …… one

Calculate determinant: A ^ n (A-1) ^ n (a-n)^n a Calculate the determinant: a^n (a-1)^n …… …… (a-n)^n a^n-1 (a-1)^n-1 …… …… (a-n)^n-1 …… …… …… …… …… a^n (a-1)^n …… …… (a-n)^n a (a-1) …… …… (a-n) 1 1 …… …… one

Exchange the last row with the previous row in turn until the first row. Use the same method to exchange the determinant into 1A... Other columns do not write a ^ n total exchange n + (n-1) +... + 1 = n (n + 1) / 2 times, exchange the same column, the sign offset D = 1... 1a-n... A... (a-N) ^ n... A ^ n -- this is Vandermonde determinant = n

Determinant calculation x y x + Y Y Y x + y x x + y x y
|x y x+y |
|y x+y x |
|x+y x y |

=X * (x + y) * y + y * x * (x + y) + y * x * (x + y) - (x + y) * (x + y) - x * X-Y * y = 3xy (x + y) - (x + y) ^ 3-x ^ 3-y ^ 3 = believe the following. A1 A2 a3B1 B2 b3c1 C2 C3 = a1b2c3 + a2b3c1 + b1c2a3-a3b2c1-a2b1c3-a1b3c2

Calculation of I x y x + y I y x + y x I x + y x y I determinant
We need to calculate by properties

c1+c2+c3
2(x+y) y x+y
2(x+y) x+y x
2(x+y) x y
r3-r2,r2-r1
2(x+y) y x+y
0 x -y
0 -y y-x
--In this case, the diagonal rule is used
= 2(x+y)[x(y-x)-y^2]
= -2(x^3+y^3).