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Prove: |(2y, y-z-x,2y),(2z,2z, z-x-y),( ...

Prove: `|(2y, y-z-x,2y),(2z,2z, z-x-y),( x-y-z,2x,2x)|=(x+y+z)^3`

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To prove that the determinant \[ D = \begin{vmatrix} 2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x \end{vmatrix} ...
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Using properties of determinants, prove that |[2y,y-z-x,2y],[2z,2z, z-x-y],[ x-y-z, 2x,2x]|=(x+y+z)^3

Prove that : |{:(x-y-z,2x,2x),(2y,y-z-x,2y),(2z,2z,z-x-y):}|

Prove that : |{:(y+z,x,y),(z+x,z,x),(x+y,y,z):}|=(x+y+z)(x-z)^(2)

Prove that : |{:(x+y+2z,x,y),(z,y+z+2x,y),(z,x,x+a+2y):}|=2(x+y+)^(3)

Prove that |{:(x,y,z),(x^2,y^2,z^2),(yz,zy,xy):}|=|{:(1,1,1),(x^2,y^2,z^2),(x^3,y^3,z^3):}|=(y-z)(z-x)(x-y)(yz+zy+xy)

Prove the following : |{:(x,x^(2),y+z),(y,y^(2),z+x),(z,z^(2),x+y):}|=(y-z)(z-x)(x-y)(x+y+z)

Prove that |{:(x^(2),,x^(2)-(y-z)^(2),,yz),(y^(2),,y^(2)-(z-x)^(2),,zx),(z^(2),,z^(2)-(x-y)^(2),,xy):}| =(x-y) (y-z) (z-x)(x+y+z) (x^(2)+y^(2)+z^(2))

Prove that : =2|{:(1,1,1),(x,y,rz),(x^(2),y^(2),z^(2)):}|=(x-y)(y-z)(z-x)

Prove that |x^2x^2-(y-z)^2y z y^2y^2-(z-x)^2z x z^2z^2-(x-y)^2x y|=(x-y)(y-z)(z-x)(x+y+z)(x^2+y^2+z^2)dot

RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Show that | (-a(b^2 + c^2 - a^2), 2b^3, 2c^3), (2a^3, -b(c^2 + a...

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  2. Prove: |(1+a,1, 1),( 1, 1+a, 1),(1, 1, 1+a)|=a^3+3a^2

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  3. Prove: |(2y, y-z-x,2y),(2z,2z, z-x-y),( x-y-z,2x,2x)|=(x+y+z)^3

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  4. |[y+z ,x, y],[ z+y, z, x],[x+y, y ,z]|=(x+y+z)(x-z)^2

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  5. Using properties of determinants, prove that |(a+x, y, z),( x, a+y, z...

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  6. Prove: |a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a)(a+b+c)

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  7. Without expanding, prove that |(a, b, c),( x, y, z),( p, q ,r)|=|(x, y...

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  8. Show that |x+1x+2x+a x+2x+3x+b x+3x+4x+c|=0 where a ,\ b ,\ c are in A...

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  9. Show that [[x-3,x-4,x-alpha],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]=0 whe...

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  10. If a ,\ b ,\ c are real numbers such that |b+cc+a a+b c+a a+bb+c a+...

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  11. a!=p , b!=q,c!=r and |(p,b,c),(a,q,c),(a,b,r)|=0 the value of p/(p-...

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  12. Show that x=2 is a root of the equation |x-6-1 2-3xx-3-3 2xx+2|=0 a...

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  13. Solve the following determinant equations: |(x+a, b, c),( a, x+b, c)...

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  14. Solve the following: |[1,x,x^2], [1,a, b^2], [1,b,c^2]|=0 , a!=b

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  15. If a ,b and c are all non-zero and |1+a1 1 11+b1 1 11+c|=0, then pro...

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  16. Find the area of the triangle with vertices A\ (5,\ 4),\ \ B(-2,\ 4...

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  17. Show that pointsA (a , b + c), B (b , c + a), C (c , a + b)are collin...

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  18. If the points (a1, b1),\ \ (a2, b2) and (a1+a2,\ b1+b2) are collinear,...

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  19. If the points (2,\ -3),\ (lambda,\ -1) and (0,\ 4) are collinear, find...

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  20. Using determinants, find the area of the triangle whose vertices are...

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