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Prove that abs[[1,x,x^2],[1,y,y^2],[1,z,...

Prove that `abs[[1,x,x^2],[1,y,y^2],[1,z,z^2]]=(x-y)(y-z)(z-x)`

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Consider the determinant Delta=abs[[x,x^2,1+x^3],[y,y^2,1+y^3],[z,z^2,1+z^3]] , Where x,y,z are different. Show that if Delta=0 ,then 1+xyz=0

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BODY BOOKS PUBLICATION-DETERMINANTS-EXERCISE
  1. Prove that abs[[1!,2!,3!],[2!,3!,4!],[3!,4!,5!]]=4!

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  2. Solve the system of Linear equations x+2y+z=8,2x+y-z=1,x-y+z=2

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  3. Inverse of the matrix [[0,1,2],[0,1,1],[1,0,2]]

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  4. Find the values of x in which abs[[3,x],[x,1]]=abs[[3,2],[4,1]]

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  5. Using the property of determinants,show that the points A(a,b+c),B(b...

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  6. Examine the consistency of system of following equations: 5x-6y+4z...

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  7. Let B is a square matrix of order 5, then abs[kB] is equal to….

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  8. Prove that abs[[1,x,x^2],[1,y,y^2],[1,z,z^2]]=(x-y)(y-z)(z-x)

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  9. Check the consistency of the following equations, 2x+3y+z=6,x+2y-z...

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  10. If abs[[x,3],[5,2]]=5,then x=…..

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  11. By using properties of determinants, prove that |[y+k,y,y],[y,y+k,y],...

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  12. Solve the following system of linear Equations,using matrix method, ...

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  13. Find values of x if i) |[2 , 4],[ 5, 1]|=|[2 x, 4], [6, x]| ii) ...

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  14. Find the values of x, if |(2,3),(4,5)|=|(x,3),(2x,5)|

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  15. Using properties of determinants prove the following. abs[[1,x,x^2...

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  16. Find the adjoint of B=[[1,-1,2],[2,3,5],[-2,0,1]]

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  17. By using properties of determinants, prove that |[x+4,2x,2x],[2x,x+4,...

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  18. Prove that abs[[y+k,y,y],[y,y+k,y],[y,y,y+k]]=k^2(3y+k)

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  19. If A = [[1,2],[4,2]] then show that |2A|= 4|A|

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  20. Prove that abs[[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=(a+b+c)^3

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