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prove that |{:(a,b,0),(0,a,b),(b,0,a):}|...

prove that `|{:(a,b,0),(0,a,b),(b,0,a):}|=a^3+b^3` , hence , find the value of the
following determinant :`|{:(2ab,a^2,b^2),(a^2,b^2,2ab),(b^2,2ab,a^2):}|`

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The correct Answer is:
`-(a^3+b^3)^2`
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prove that , |{:(2ab,a^2,b^2),(a^2,b^2,2ab),(b^2,2ab,a^2):}|=-(a^3+b^3)^2

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Using properties of determinants , prove that, |{:(a,b,c),(b,c,a),(c,a,b):}|=-(a^3+b^3+c^3-3abc) and hence show that, |{:(2bc-a^2," "c^2," "b^2),(" "c^2,2ca-b^2," "a^2),(" "b^2," "a^2,2ab-c^2):}|=(a^3+b^3+c^3-3abc)^2

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The value of the determinant |(1+a^(2)-b^(2),2ab,-2b),(2ab,1-a^(2)+b^(2),2a),(2b,-2a,1-a^(2)-b^(2))| is equal to

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CHHAYA PUBLICATION-DETERMINANT -EXERCISE 2B
  1. Evaluate |{:(0,c,b),(c,0,a),(b,a,0):}|, hence show that . |{:(0,c,b)...

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  2. Express the square of |{:(a1,b1,0),(a2,b2,0),(a3,b3,0):}| as a third o...

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  3. prove that |{:(a,b,0),(0,a,b),(b,0,a):}|=a^3+b^3 , hence , find the va...

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  4. Prove that , the value of the determinant |{:(x+1,x+2,x+4),(x+3,x+5,...

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  5. |{:(b+c,a-c,a-b),(b-c,c+a,b-a),(c-b,c-a,a+b):}|=8abc

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  6. |{:(" "a^2," "bc,c^2+ca),(a^2+ab," "b^2," "ca),(" "ab,b^2+bc," "c^2...

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  7. |{:(2costheta,1,0),(1,2costheta,1),(0,1,2costheta):}|=(sin4theta)/(sin...

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  8. |{:(a,b,ax+by),(b,c,bx+cy),(ax+by,bx+cy,0):}|=(b^2-ac)(ax^2+2bxy+cy^2)

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  9. |{:(3x^2,3x,1),(x^2+2x,2x+1,1),(2x+1,x+2,1):}|=(x-1)^3

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  10. |{:(x+y+z," "-z," "-y),(" "-z,x+y+z," "-x),(" "-y," "-x...

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  11. |{:(1,1,1),(""^(n)c1,""^(n+1)c1,""^(n+2)c1),(""^(n+1)c2,""^(n+2)c2,""^...

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  12. |{:(x+y+z," "-z," "-y),(" "-z,x+y+z," "-x),(" "-y," "-x...

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  13. |{:(cos(x+y),sin(x+y),-cos(x+y)),(sin(x-y),cos(x-y),sin(x-y)),(sin2x,0...

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  14. prove that, |{:(0,cosalpha,-sinalpha),(sinalpha,0,cosalpha),(cosalpha,...

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  15. If a ,b ,c are in A.P , show that , |{:(x+1,x+2,x+a),(x+2,x+3,x+b),(x+...

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  16. If A,B,C , are the angles of a triangles, show that, |{:(sin^2A,coaA,1...

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  17. without expanding prove that [22-24] |{:(7,12,-3),(9,14,-1),(8,13,-2...

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  18. |{:(1,bc,bc(b+c)),(1,ca,ca(c+a)),(1,ab,ab(a+b)):}|=0

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  19. |{:(logx xyz,logxy,logxz),(logyxyz,1,logyz),(logzxyz,logzy,1):}|=0

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  20. if |{:(a+ib,c+id),(-c+id,a-ib):}|xx|{:(alpha-ibeta,gamma-idelta),(-gam...

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