求证下面的行列式等于0｜a^2(a+1)^2(a+2)^2(a+3)^2｜｜b^2(b+1)^2(b+2)^2(b+3)^2｜｜c^2(c+1)^2(c+2)^2(c+3)^2｜｜d^2(d+1)^2(d+2)^2(d+3)^2｜

Question

求证下面的行列式等于0
｜a^2 (a+1)^2 (a+2)^2 (a+3)^2｜
｜b^2 (b+1)^2 (b+2)^2 (b+3)^2｜
｜c^2 (c+1)^2 (c+2)^2 (c+3)^2｜
｜d^2 (d+1)^2 (d+2)^2 (d+3)^2｜

Answer 1

第一列-3×第二列+3×第三列-第四列=0.
恒等式：x²-3(x+1)²+3(x+2)²-(x+3)²=0.
x²-3(x+1)²+3(x+2)²-(x+3)²
=x²-(x+3)²+3(x+2)²-3(x+1)²
=(x-x-3)(x+x+3)+3(x+2-x-1)(x+2+x+1)
=-3(2x+3)+3(2x+3)=0.