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evaluate: |(a+x,y,z),(x,a+y,z),(x,y,a+z)...

evaluate: `|(a+x,y,z),(x,a+y,z),(x,y,a+z)|`

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The correct Answer is:
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We have `|(a+x, y, z),(x, a+y, z),(x, y, a+z)|=|(a, -a, 0),(0,a,-a),(x,y,a+z)|`
`[ ( :'R_(1)toR_(1)-R_(2)), "and"(R_(2)toR_(2)-R_(3))]`
`|(a,0,0),(0,a,-a),(x,x+y,a+z)|` `[ :' C_(2)toC_(2)+C_(1)]`
`=a(a^(2)+az+ax+ay)`
`=a^(2)(a+z+x+y)`
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NCERT EXEMPLAR-DETERMINANTS-Determinants
  1. evaluate: |(x^(2)-x+1, x-1),(x+1,x+1)|

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  2. evaluate: |(a+x,y,z),(x,a+y,z),(x,y,a+z)|

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  3. evaluate: |(0,xy^(2),xz^(2)),(x^(2)y,0,yz^(2)),(x^(2)z,zy^(2),0)|

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  4. evaluate: |(3x,-x+y,-x+z),(x-y,3y,z-y),(x-z,y-z,3z)|

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  5. evaluate: |(x+4,x,x),(x,x+4,x),(x,x,x+4)|

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  6. evaluate: |(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)|

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  7. prove that:|(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y...

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  8. prove that:|(y+z,z,y),(z,z+x,x),(y,x,x+y)|=4xyz

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  9. Using properties of determinants, prove that 3 2 (a 1) 3 3 1 2a 1...

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  10. If A+B+C=0, then prove that Det[[1,cosC,cosB],[cosC,1,cosA],[cosB,cosA...

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  11. If the coordinates of the vertices of an equilateral triangle with sid...

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  12. Find the value of theta if |[1,1,sin 3theta] , [-4,3,cos 2theta] , [7,...

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  13. If |[4-x, 4+x, 4+x], [4+x, 4-x, 4+x],[4+x, 4+x, 4-x]| = 0 find the va...

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  14. If a(1),a(2),a(3),….,a(r) are in GP, then prove that the determinant |...

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  15. Show that the points (a+5,a-4),(a-2,a+3) and (a,a) do not lie on a str...

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  16. Show that DeltaABC is an isosceles triangle, if the determinant Delt...

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  17. Find A^(-1) if A=|(0,1,1),(1,0,1),(1,1,0)| and show that A^(-1)=(A^(2)...

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  18. If A\|(1,2,0),(-2,-1,-2),(0,-1,1)|, then find the value of A^(-1) Us...

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  19. Using matrix method, solve the system of equation 3x+2y-2z=3, x+2y+3z=...

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  20. If A=|(2,2,-4),(-4,2,-4),(2,-1,5)| and B=|(1,-1,0),(2,3,4),(0,1,2)| th...

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