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(x+y+z) is fator of : {:|(x-y-z,2x,2x)...

(x+y+z) is fator of :
`{:|(x-y-z,2x,2x),(2y,y-z-x,2y),(2z,2z,z-x-y)|`

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Show that: |[x-y-z,2x,2x],[2y,y-z-x,2y],[2z,2z,z-x-y]|=(x+y+z)^3

If x,y,z are different and Delta= {:|(x,x^2,1+x^3),(y,y^2,1+y^3),(z,z^2,1+z^3)|=0 , show that xyz=-1

By using properties of determinants, show that : |[x+y+2z,x,y],[z,y+z+2x,y],[z,z,z+x+2y]| = 2(x+y+z)^3

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)| = (x-y)(y-z)(z-x)

Prove the following identities : |{:(x,x^(2),x^(3)),(y,y^(2),y^(3)),(z,z^(2),z^(3)):}|=xyz(x-y)(y-z)(z-x) .

If x, y, z are all distinct and |(x,x^(2),1+x^(3)),(y,y^(2),1+y^(3)),(z,z^(2),1+z^(3))|=0 then value of x y z is :

Using matrices, solve the following system of equations for x,y and z: {:(x+2y+z=8),(2x+y-z=1),(x-y+z=2):}

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MODERN PUBLICATION-DETERMINANTS-EXERCISE
  1. Without expanding, prove that : |{:(1, bc, a(b+ c) ),(1, ca, b ( c+ a)...

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  2. Without expanding the determinant, show that : (frac{1}{a}+frac{1}{b}+...

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  3. (x+y+z) is fator of : {:|(x-y-z,2x,2x),(2y,y-z-x,2y),(2z,2z,z-x-y)|

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  4. Without expanding the determinant, prove that |[a,a^2,bc],[b,b^2,ca],[...

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  5. Evaluate the following: {:|(b+c,a,a),(b,c+a,b),(c,c,a+b)|

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  6. Evaluate |(x+lamda,x,x),(x,x+lamda,x),(x,x,x+lamda)|

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  7. Evaluate the following: {:|(1,a,b+c),(1,b,c+a),(1,c,a+b)|

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  8. Evaluate |[x,y,x+y],[y,x+y,x],[x+y,x,y]|

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  9. Evaluate |[1,x,y],[1,x+y,y],[1,x,x+y]|

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  10. Using the property of determinants and without expanding , prove that:...

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  11. Using the property of determinants and without expanding , prove that:...

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

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  13. using properties of determinant, prove that abs{:(y+k , y , y ),(y , y...

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  14. Prove that: {:|(x,y,x+y),(y,x+y,x),(x+y,x,y)| = -2(x^3+y^3)

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  15. By using properties of determinants, show that : |[1,a,a^2],[1,b,b^2]...

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  16. Prove that: {:|(a,b-c,c+b),(a+c,b,c-a),(a-b,b+a,)|=(a+b)+c)(a^2+b^2+...

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  17. Prove that: {:|(1,a,a),(a,1,a),(a,a,1)|= (2a+1),(1-a)^2

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  18. Prove that: {:|(1,x,x^3),(1,y,y^3),(1,z,z^3)| = (x-y)(y-z)(z-x)(x+y+...

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  19. Prove that: {:|(1,1,1),(a,b,c),(bc,ca,ab)| = (a-b)(b-c)(c-a)

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  20. Prove that: {:|(1,a,bc),(1,b,ca),(1,c,ab)|=(a-b)(b-c)(c-a)

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