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Prove that the following. [[b+c,a,a],[b,...

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

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Prove that the following. [[b^2+c^2,ab,ac],[ab,c^2+a^2,bc],[ca,cb,a^2+b^2]]=4a^2b^2c^2

Prove the following: [[b+c,a+b,a],[c+a,b+c,b],[a+b,c+a,c]] = a^3+b^3+c^3-3abc

Prove that the following. [[1,1,1],[b+c,c+a,c+a],[b^2+c^2,c^2+a^2,a^2+b^2]] =(b-c)(c-a)(a-b)

Prove the following: [[-a^2,ab,ac],[ab,-b^2,bc],[ac,bc,-c^2]]=4a^2b^2c^2

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

Prove the following: [[(b+c)^2,a^2,bc],[(c+a)^2,b^2,ca],[(a+b)^2,c^2,ab]] = (a^2+b^2+c^2)(a+b+c)(b-c)(c-a)(a-b)

MODERN PUBLICATION-DETERMINANTS-PROBLEM
  1. The value of [[1,2,3],[3,5,2],[8,14,20]] is .

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  2. Solve the following : [[x+1,omega,omega^2],[omega,x+omega^2,1],[omega^...

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

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  4. Solve : [[7,6,x],[2,x,2],[x,3,7]] = 0

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  5. Solve the following : [[1+x,1,1],[1,1+x,1],[1,1,1+x]]=0

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

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  7. Prove that the following. [[a+d,a+d+k,a+d+c],[c,c+b,c],[d,d+k,d+c]]=ab...

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  8. 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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  9. 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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  10. 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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  11. 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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  12. Prove that [[1,a,a^2 - bc],[1,b,b^2 - ca],[1,c,c^2 - ab]]

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  13. 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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  14. 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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  15. Factorize the following. [[a,b,c],[b+c,c+a,a+b],[a^2,b^2,c^2]]

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

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

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  18. 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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  19. 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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  20. Show that the homogenous system of equations x-2y+z = 0 x+y-z = 0 ...

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