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Without expanding prove that |[1,a,a^(2)...

Without expanding prove that `|[1,a,a^(2),-bc],[1,b,b^(2),-ca],[1,c,c^(2),-ab]|=0`

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Prove that [[1,a,a^2 - bc],[1,b,b^2 - ca],[1,c,c^2 - ab]]

Show that without expanding at any stage |{:(1/a,a^2,bc),(1/b,b^2,ca),(1/c,c^2,ab):}|=0

Without expanding the determinants prove that |{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}| = |{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}|

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))

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]|

Evaluate [[a,a^2-bc,1],[b,b^2-ac,1],[c,c^2-ab,1]]

Without expanding show that |{:(1,a,b+c),(1,b,c+a),(1,c,a+b):}|=0

Prove the following : [[1,bc,a(b+c)],[1,ca,b(c+a)],[1,ab,c(a+b)]] =0

Show that: abs((1,a,a^2),(1,b,b^2),(1,c,c^2))=(a-b)(b-c)(c-a)

SHARAM PUBLICATION-DETERMINANT-EXAMPLE
  1. Evaluate the following determinants: [[0,a^2,b],[b^2,0,a^2],[a,b^2,0...

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  2. 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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  3. Without expanding prove that |[1,a,a^(2),-bc],[1,b,b^(2),-ca],[1,c,c^(...

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  4. Prove the following : [[1,bc,a(b+c)],[1,ca,b(c+a)],[1,ab,c(a+b)]]=0

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  5. Given the equations x=cy+bz, y=az+cx and z=bx+ay where x,y and z a...

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  6. Eliminate x,y,z from a=x/y-z, b=y/z-x, c=z/x-y

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  7. Prove that abs[[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]]=2abs[[a,b,c]...

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

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  9. Prove that |[x+y, x, x],[5x+4y, 4x, 2x], [10x+8y, 8x, 3x]|=x^3

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  10. Using properties of determinants prove that |{:(a+x,y,z),(x,a+y,z),...

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  11. Prove that the following. [[b+c,a,a],[b,c+a,b],[c,c,a+b]]=4ab

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  12. Prove the following: [[1,1,1],[a,b,c],[a^3,b^3,c^3]] =(b-c)(c-a)(a...

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  13. Prove that |[x, y, z],[x^2, y^2, z^2], [x^3, y^3, z^3]|=xyz(x-y)(y-z)(...

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  14. Prove that the following. [[a,b,c],[a^2,b^2,c^2],[bc,ca,ab]]=(b-c)(c-a...

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  15. Prove the following: [[(b+c)^2,a^2,bc],[(c+a)^2,b^2,ca],[(a+b)^2,c^2...

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  16. Using the properties of determinants, show that abs[[1+a^2-b^2,2ab,-2b...

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  17. Prove that: |[a, b, c],[a-b, b-c, c-a ], [b+c, c+a, a+b]|= a^3+b^3+c^3...

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  18. If [[x,x^2,x^3-1],[y,y^2,y^3-1],[z,z^2,z^3-1]]=0 then prove that xyz...

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  19. Using properties of the determinants, prove that: |[2y, y-z-x, 2y],[2z...

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  20. Prove that: |[1, 1+p, 1+p+q],[2, 3+2p, 1+3p+2p], [3, 6+3p, 1+6p+3q ]|=...

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