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Using properties of determinants, show...

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

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Using properties of determinant show that: |[1 , a , bc] , [1 , b , ca] , [1 , c , a b]|=(a-b)(b-c)(c-a)

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

Using properties of determinants, prove the following: |[a^2 + 1,ab, ac], [ab,b^2 + 1,b c],[ca, cb, c^2+1]|=1+a^2+b^2+c^2

Using properties of determinant, prove that |(2a, a-b-c, 2a), (2b, 2b, b-c-a), (c-a-b,2c,2c)|=(a+b+c)^(3) .

Using the property of determinants and without expanding, prove that: |-a^2a b a c b a b^2b cc a c b-c^2|=4a^2b^2c^2

Using properties of determinant prove that |(a^(2)+1, ab, ac),(ab, b^(2)+1, bc),(ca, cb,c^(2)+1)|=(1+a^(2)+b^(2)+c^(2)) .

Without expanding, show that the value of each of the determinants is zero: |1/a a^2 bc 1/b b^2 ac 1/c c^2 ab|

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Using properties of determinants, prove the following : |[a, a^2,bc],[b,b^2,ca],[c,c^2,ab]|=(a-b)(b-c)(c-a)(b c+c a+a b)

RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. If x!=y!=z and |x x^2 1+x^3 y y^2 1+y^3 z z^2 1+z^3|=0 , then prove th...

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

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

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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, 0,a] , [x,a-1,a], [x^2a, x, a]| , using properties of ...

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  7. Show that: x p q p x q q q xx-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+yx x5x+4y4x2x 10 x+8y8x3x|=x^3 .

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  11. Show that: |[1 ,1+p,1+p+q],[2, 3+2p,1+3p+2q],[3, 6+3p,1+6p+3q]|=1 .

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  12. Show that: |[a, a+b ,a+b+c],[2a,3a+2b,4a+3b+2c],[3a,6a+3b, 10 a+6b+3c]...

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  13. Show that: |[b+c,c+a ,a+b],[ q+r, r+p, p+q],[ y+z ,z+x,x+y]|=2|[a, b, ...

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  14. Prove that: determinant \begin{vmatrix} 1+ a & 1 & 1 \\ 1 & 1+b...

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

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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. Show that: |(b+c)^2b a c a a b(c+a)^2c b a c b c(a+b)^2|=2a b c(a+b+c)...

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  18. 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^...

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  19. Show that: |b^2+c^2a b a c b a c^2+a^2b cc a c b a^2+b^2|=4a^2\ b^2\ c...

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  20. Prove that: |a b a x+b y b c b x+c y a x+b y b x+c y0|=(b^2-a c)(a x^2...

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