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Without expanding, prove the following ...

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

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MODERN PUBLICATION-DETERMINANTS-EXERCISE
  1. 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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  2. By using properties of determinants, show that : |[x+y+2z,x,y],[z,y+...

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

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

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

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

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

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  8. 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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  9. 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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  10. 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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  11. Using properties of determinants, prove that: |[3a,-a+b,-a+c],[-b+a,3b...

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  12. 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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  13. 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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  14. Prove that the determinant |[x,sintheta,costheta],[-sintheta,-x,1],[co...

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

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

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  17. If a, b, c are positive and unequal, show that value of the determinan...

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  18. Without expanding show that following : |[a,a+b,a+b+c],[2a,3a+2b,4a+3b...

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  19. Prove that: |[1,1+p,1+p+q],[2,3+2p,4+3p+2q],[3,6+3p,10+6p+3q]|=1

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

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