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Given the equations x=cy+bz, y=az+cx a...

Given the equations
x=cy+bz, y=az+cx and z=bx+ay
where x,y and z are not all zero, prove that `a^2+b^2+c^2+2abc=1` by determinant method.

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

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

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

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

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

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

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

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

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

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

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

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

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  15. 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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  16. 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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  17. Using properties of the determinants, prove that: |[2y, y-z-x, 2y],[2z...

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

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  19. Show that: |[x, y ,z],[x^2, y^2, z^2], [yz, zx, xy ]|=(x-y)(y-z)(z-x)....

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  20. Prove [[a^3-x^3,a^2,a],[b^3-x^3,b^2,b],[c^3-x^3,c^2,c]] = (a-b)(a-c)(b...

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