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汽车分级 a b c d
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线性代数问题证明:|1 1 1 1 ||a b c d ||a² b² c² d² |=(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)(a+b+c+d)|a4 b4 c4 d4 |本人不会打四次方符号!两边是1条直线,问提应该看得懂吧!
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用初等变换把它化成阶梯矩阵:1.第二行减去第一行的a倍;第三行减去第一行的a^2倍;第四行减去第一行的a^3倍,得:1 1 1 10 b-a c-a d-a0 b2-a2 c2-a2 d2-a20 b4-a4 c4-a4 d4-a42.第三行减去第二行的(b+a)倍;第四行减去第二行的(b2+a2)(b+a)倍,得:1 1 1 10 b-a c-a d-a0 0 (c-a)(c-b) (d-a)(d-b)0 0 x1 y1其中,x1=(c-a)[(c2+a2)(c+a)-(b2+a2)(b+a)]=(c-a)(c-b)(a2+b2+c2+ab+ac+bc)y1=(d-a)[(d2+a2)(d+a)-(b2+a2)(b+a)]=(d-a)(d-b)(a2+b2+d2+ab+ad+bd)3.第四行减去第三行的(a2+b2+c2+ab+ac+bc)倍,得:1 1 1 10 b-a c-a d-a0 0 (c-a)(c-b) (d-a)(d-b)0 0 0 y2其中,y2=(d-a)(d-b)(d-c)(a+b+c+d)4.所以,行列式=对角线上的数相乘=(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)(a+b+c+d)
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COPYRIGHT (C) 1985 BY DIGITAL EQUIPMENT CORPORATION
ALL RIGHTS RESERVED
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