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Prove without expanding that |[bc,a,a^2]...

Prove without expanding that `|[bc,a,a^2],[ca,b,b^2],[ab,c,c^2]|=|[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|`

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Without expanding prove abs((bc, a, a^2),(ca, b, b^2),(ab, c, c^2))=abs((1, a^2, a^3),(1, b^2, b^3),(1, c^2, c^3))

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SHARAM PUBLICATION-DETERMINANT-EXAMPLE
  1. Solve the following : [[0,x-a,x-b],[x+a,0,x-c],[x+b,x+c,0]]=0

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  2. Prove that the following. [[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=...

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  3. Prove without expanding that |[bc,a,a^2],[ca,b,b^2],[ab,c,c^2]|=|[1,a^...

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  4. Factorise the determinant |[x^3-a^3,x^2,x],[b^2-a^3,b^2,b],[c^3-a^3,c^...

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  5. Show that (a+1) is a factor of abs([(a+1),2,3],[1,a+1,3],[3,-6,a+1])

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  6. Prove that the following. [[a,b,c],[x,y,z],[p,q,r]]=[[y,b,q],[x,a,p],[...

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  7. Prove the following: [[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]] =1+...

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  8. Solve for x, |{:(15-2x,11,10),(11-3x,17,16),(7-x,14,13):}|=0

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  9. What is the value of |[i^(103), 3 ,i^(101)],[i^(56), 5, i^(54)],[i^(23...

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  10. Without expanding find the value of the deteminant |[3, 6, 9],[-2, 4 ,...

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  11. Prove that:|[1, a ,a^2],[a^2, 1 ,a],[a, a^2, 1]| is a perfect square.

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  12. Without expanding prove that: |[12, 2, 4, -5, 1],[-8, 1, -5, 2, -1],[6...

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  13. Prove that: |[1, 1, 1],[a, b, c],[a^2, b^2, c^2]|=(a-b)(b-c)(c-a)

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  14. Solve : |[x-a, 0, 0],[a, x-b, 0],[a, b, x-c]|=0

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  15. What is the value of the determinant |[0, 0, 2],[0, 2, 0],[2, 0 ,0]|

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  16. Solve the following : [[1+x,1,1],[1,1+x,1],[1,1,1+x]]=0

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  17. Find the value of |{:(17,58,97),(19,60,99),(18,59,98):}| without expan...

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  18. Prove that the following. [[1+a,1,1],[1,1+b,1],[1,1,1+c]] = abc(1+1/...

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  19. Evaluate the following determinants: [[0,a^2,b],[b^2,0,a^2],[a,b^2,0...

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  20. Prove that: |[a+b+2c, a, b],[c, b+c+2a, b],[c, a, c+a+2b]|=2(a+b+c)^3

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