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Without expanding, prove the following ...

Without expanding, prove the following
`|(a,b,c),(a^2,b^2,c^2),(bc,ca,ab)|=(a-b)(b-c)(c-a)(ab+bc+ca)`

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MODERN PUBLICATION-DETERMINANTS-EXERCISE
  1. Prove that: {:|(1,a,bc),(1,b,ca),(1,c,ab)|=(a-b)(b-c)(c-a)

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

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  3. Without expanding, prove the following |(a,b,c),(a^2,b^2,c^2),(bc,c...

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  4. Prove that: {:|(a^2,a,b+c),(b^2,b,c+a),(c^2,c,ab)| = -(a+b+c)(a-b)(b...

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  5. Prove that: {:|(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c)|=3abc -a^3-b^3-c^...

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  6. Prove that |{:(b^(2)+c^(2),ab,ac),(ab,c^(2)+a^(2),bc),(ac,bc,a^(2)+b...

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  7. Prove that: {:|(1+a^2-b^2,2ab,-2b),(2ab,1-a^2+b^2,2a),(2b,-2a,1-a^2-...

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  8. Prove that: {:|(x,y,z),(x^2,y^2,z^2),(x^3,y^3,z^3)|=|(x,x^2,x^3),(y,...

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  9. Prove that: |[x,x^2,yz],[y,y^2,zx],[z,z^2,xy]|=(x-y)(y-z)(z-x)(xy+yz+z...

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  10. By using properties of determinants, show that : |[x+y+2z,x,y],[z,y+...

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  11. Without expanding, prove the following |(b+c,c+a,a+b),(c+a,a+b,b+c)...

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  12. Without expanding, prove the following |(x,x+y,x+2y),(x+2y,x,x+y),(...

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  13. Prove that |{:(b+c,a,a),(b,c+a,b),(c,c,a+b):}| = 4 abc

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  14. Prove that |[1,x,x^2-yz],[1,y,y^2-zx],[1,z,z^2-xy]|= 0

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  15. Using properties of determinants, prove that: |[x,x^2,1+px^3],[y,y^2,1...

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  16. Prove that : |[x+y+z,-z,-y],[-z, x+y+z, -x],[-y,-x,x+y+z]|= 2(x+y)(y+z...

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  17. Show that: |[x-y-z,2x,2x],[2y,y-z-x,2y],[2z,2z,z-x-y]|=(x+y+z)^3

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  18. Show that: |[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]|=(a+b+c)^3

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  19. Using properties of determinants, prove that: |[3a,-a+b,-a+c],[-b+a,3b...

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  20. Solve for x":"|(a+x,a-x,a-x),(a-x,a+x,a-x),(a-x,a-x,a+x)|=0

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