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

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

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

Prove that |(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)| is divisible by a+b+c. Also find the value of the quotient.

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MODERN PUBLICATION-DETERMINANTS-EXERCISE
  1. 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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  2. Prove that |{:(b^(2)+c^(2),ab,ac),(ab,c^(2)+a^(2),bc),(ac,bc,a^(2)+b...

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  3. 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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  4. 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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  5. 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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  6. By using properties of determinants, show that : |[x+y+2z,x,y],[z,y+...

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

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

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

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

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

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  12. 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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  13. 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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  14. 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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  15. Using properties of determinants, prove that: |[3a,-a+b,-a+c],[-b+a,3b...

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  16. 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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  17. Solve the equation |(x-2,2x-3,3x-4),(x-4,2x-9,3x-16),(x-8,2x-27,3x-64)...

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  18. Prove that the determinant |[x,sintheta,costheta],[-sintheta,-x,1],[co...

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  19. Using properties of determinants, show that: |[[(b+c)^2, a^2, a^2],[b^...

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  20. Prove that |((b+c)^2,ab,ca),(ab,(a+c)^2,bc),(ac,bc,(a+b)^2)|=2abc(a+b...

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