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

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

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MODERN PUBLICATION-DETERMINANTS-PROBLEM
  1. Prove that the following. [[1,1,1],[b+c,c+a,c+a],[b^2+c^2,c^2+a^2,a^2+...

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  2. Prove that [[b^2c^2, bc,b+c],[c^2a^2,ca,c+a],[a^2b^2,ab,a+b]] = 0

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

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  4. Show that |{:((b+c)^2,a^2,a^2),(b^2,(c+a)^2,b^2),(c^2,c^2,(a+b)^2):}...

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

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  6. Prove that the following. [[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]]=...

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  7. Prove that the following. [[1,1,1],[b+c,c+a,c+a],[b^2+c^2,c^2+a^2,a^2+...

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  8. Factorize the following. [[a,b,c],[b+c,c+a,a+b],[a^2,b^2,c^2]]

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  9. Prove the following : [[1,x,x^2],[x^2,1,x],[x,x^2,1]]=(1-x^3)^2

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  10. Prove that the following. [[b^2+c^2,ab,ac],[ab,c^2+a^2,bc],[ca,cb,a^2+...

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  11. Prove that [[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]] = 1+a^2+b^2+c...

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  12. Show that [[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]] = 3(a+b+c)(ab...

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  13. Show that the homogenous system of equations x-2y+z = 0 x+y-z = 0 ...

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  14. Prove that the following. [[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=...

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  15. Examining consistency and solvability, solve the following equation by...

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

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  17. Prove the following: [[a+b+c,-c,-b],[-c,a+b+c,-a],[-b,-a,a+b+c]] =...

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  18. Prove the following: |[ax-by-cz,ay+bx,az+cx],[bx+ay,by-cz-ax,bz+cy],...

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  19. 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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  20. If x,y,z are positive and are the pth, qth and rth terms of a G.P. the...

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