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If Dr=|2^(r-1)2(3^(r-1))4(5^(r-1))x y...

If `D_r=|2^(r-1)2(3^(r-1))4(5^(r-1))x y z2^n-1 3^n-1 5^n-1|` then prove that `sum_(r=1)^n D_r=0.`

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ALLEN -DETERMINANTS-All Questions
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  2. For a fixed positive integer n , if =|n !(n+1)!(n+2)!(n+1)!(n+2)!(n+3)...

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  3. If Dr=|2^(r-1)2(3^(r-1))4(5^(r-1))x y z2^n-1 3^n-1 5^n-1| then p...

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  10. Let alpha1,alpha2a n d\ beta1,beta2 be the roots of a x^2+c=0\ a...

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  11. Prove that: {:|(1,a, a^2-bc), (1,b,b^2-ca),(1,c,c^2-ab)|:}=0

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  12. If Sr = alpha^r + beta^r + gamma^r then show that |[S0,S1,S2],[S1,S2,S...

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  13. Prove that: {:|(1,a, a^2-bc), (1,b,b^2-ca),(1,c,c^2-ab)|:}=0

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  14. If u=a x^2+2b x y+c y^2,\ y^(prime)=a^('2)+2b^(prime)x y+c^(prime)y^2,...

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  15. If the system of linear equations x+a y+a z=0,\ x+b y+b z=0,\ x+c y...

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  16. If x,y and y are not all zero and if ax+by+cz=0,bx+cy+az=0 and cx...

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  17. Prove that the system of equations in x\ a n d\ y ; a x(h y)+g=0,\ h x...

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  18. If a^2+b^2+c^2=-2a n df(x)= |1+a^2x(1+b^2)x(1+c^2)x(1+a^2)x1+b^2x(1+c...

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  19. The system of equations ax+y+z=a-1,x+ay+z=a-1andx+y+az=a-1 does not h...

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  20. If D=|[1, 1, 1], [1, 1+x, 1], [1, 1, 1+y]| for x!=0, y!=0 then D is (1...

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