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Using properties of determinants, show that `|1a a^2-b c1bb^2-c a1cc^2-a b|=0`

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Prove that: |1a a^2-b c1bb^2-c a1cc^2-a b|=0

By using properties of determinants. Show that: (i) |1a a^2 1bb^2 1cc^2|=(a-b)(b-c)(c-a) (ii) |1 1 1a b c a^3b^3c^3|=(a-b)(b-c)(c-a)(a+b+c)

Using properties of determinant show that: det[[1,a,-bc1,b,-ca1,c,-ab]]=(a-b)(b-c)(c-a)det[[1,b,-ca1,c,-ab]]=(a-b)(b-c)(c-a)

By using properties of determinants. Show that: (i) |a-b-c2a2a2bb-c-a2b2c2cc-a-b|=(a+b+c)^3 (ii) |x+y+2z x y z y+z+2x y z x z+x+2y|=2(x+y+z)^3

By using properties of determinants. Show that: |1+a^2-b^2 2a b-2b2a b1-a^2+b^2 2a2b-2a1-a^2-b^2|=(1+a^2+b^2)^3

Using properties of determinants, show that |(b+a,a,a),(b,c+a,b),(c,c,a+b)|=4abc

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

Using properties of determinant prove that |a+b+c-c-b-c a+b+c-a-b-a a+b+c|=2(a+b)(b+c)(c+a)

Using properties of determinants, prove that |[a,b,c] , [a^2,b^2,c^2] , [b+c,c+a,a+b]|=(a+b+c)(a-b)(b-c)(c-a)

By using properties of determinants.Show that: det[[a^(2)+1,ab,acab,b^(2)+1,bcca,cb,c^(2)+1]]=(1+a^(2)+b^(2)+b^(2))

RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. If x!=y!=z and |xx^2 1+x^3y y^2 1+y^3z z^2 1+z^3|=0 , then prove that ...

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  2. For any scalar p prove that =|xx^2 1+p x^3y y^2 1+p y^3z z^2 1+p z^3|=...

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  3. Using properties of determinants, show that |1a a^2-b c1bb^2-c a1cc...

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  4. Prove that: |a^2+2a2a+1 1 2a+1a+2 1 3 3 1|=(a-1)^3 .

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  5. Let a ,\ b and c denote the sides B C ,\ C A and A B respectively of ...

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  6. If f(x)=|a-1 0a x a-1a x^2a x a| , using properties of determinants...

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  7. Show that: |x p q p x q q q x|=(x-p)(x^2+p x-2q^2) .

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  8. If m in N and mgeq2 prove that: |1 1 1\ ^m C1\ ^(m+1)C1\ ^(m+2)C1\ ^m...

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  9. Evaluate: =|10 ! 11 ! 12 ! 11 ! 12 ! 13 ! 12 ! 13 ! 14 !| .

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  10. Prove that: |x+y xx5x+4y4x2x 10 x+8y8x3x|=x^3 .

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  11. Show that: |1 1+p1+p+q2 3+2p1+3p+2q3 6+3p1+6p+3q|=1 .

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  12. Show that: |a a+b a+b+c2a3a+2b4a+3b+2c3a6a+3b 10 a+6b+3c|=a^3 .

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  13. Show that: |b+cc+a a+b q+r r+p p+q y+z z+xx+y|=2|a b c p q r x y z| .

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  14. Prove that: |1+a1 1 1 1+b1 1 1 1+c|=a b c(1+1/a+1/b+1/c)=a b c+b c+c a...

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  15. If a ,\ b ,\ c are the roots of the equation x^3+p x+q=0 , then fin...

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  16. Prove that: |(b+c)^2a^2a^2b^2(c+a)^2b^2c^2c^2(a+b)^2|=2a b c(a+b+c)^3

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  17. Prove that: |(a,b, ax+by),(b,c,bx+cy), (ax+by, bx+cy,0)|=(b^2-a c)(a x...

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  18. Without expanding the determinant, show that (a+b+c) is a factor of...

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  19. If a ,\ b ,\ c are roots of the equation x^3+p x^{2}+q=0 , prove tha...

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

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