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Without expanding, show that the value o...

Without expanding, show that the value of each of the determinants is zero: `|(1/a, a^2,bc),(1/b,b^2,ac) ,(1/c,c^2,ab)|`

Text Solution

Verified by Experts

Given
`|(1/a, a^2,bc),(1/b,b^2,ac) ,(1/c,c^2,ab)|`
On multiplying `R_1`, `R_2`, and `R_3` with a, b and c
= `|(1, a^3,abc),(1,b^3,abc) ,(1,c^3,abc)|`
On taking, abc common from `C_3`
= abc`|(1, a^3,1),(1,b^3,1) ,(1,c^3,1)|`
...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
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  2. Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c...

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  3. Without expanding, show that the value of each of the determinants is ...

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  4. If m is a positive integer and Dr=|2r-1\ ^m Cr1m^2-1 2^m m+1s in^2(...

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  5. Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y...

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  6. Let Deltar=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3...

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  7. If Deltar=|[2^(r-1)2 , 3^(r-1) , 4. 5^(r-1)] , [x , y , z] , [2^n-1 , ...

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  8. Find a quadratic polynomial phi(x) whose zeros are the maximum and mi...

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  10. Show that: |(a, b-c,c+b), (a+c, b, c-a), (a-b,b+a, c)|=(a+b+c)(a^2+b^2...

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  11. If a , b , c are real numbers, prove that |a b c b c a c a b|=-(a+b+c)...

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  12. Solve: |(a+x, a-x, a-x),(a-x, a+x, a-x),(a-x, a-x, a+x)|=0

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  13. Show that: |(3a,-a+b,-a+c),(-b+a,3b,-b+c),(-c+a,-c+b,3c)|=3(a+b+c)(a b...

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  14. If a,b,c are all distinct and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-...

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  15. Solve : |(x-2, 2x-3, 3x-4), (x-4, 2x-9, 3x-16) ,(x-8 ,2x-27,3x-64)|=0

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  16. If x+y+z=0 , prove that |[xa, yb,zc],[yc, za, xb], [zb, xc, ya]|=x y ...

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  17. If a , b , c are all positive and are p t h ,q t ha n dr t h terms of ...

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  18. Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-...

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  19. |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|...

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