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|{:(b+c,a,1),(c+a,b,1),(a+b,c,1):}|=0...

`|{:(b+c,a,1),(c+a,b,1),(a+b,c,1):}|=0`

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|{:(1,a,b+c),(1,b,c+a),(1,c,a+b):}|=0

prove that : |{:(1,a,b),(-a,1,c),(-b,-c,1):}|=1+a^2+b^2+c^2

|{:(1,1,1),(a,b,c),(bc,ca,ab):}|=(a-b)(b-c)(c-a)

If |{:(a,b,1),(b,c,1),(c,a,1):}|=2010 and if |{:(c-a,c-b,ab),(a-b,a-c,bc),(b-c,b-a,ca):}|-|{:(c-a,c-b,c^(2)),(a-b,a-c,a^(2)),(b-c,b-a,b^(2)):}|=p , then the number of positive divisors of p is

|{:(a-b,1,a),(b-c,1,b),(c-a,1,c):}|=|{:(a,1,b),(b,1,c),(c,1,a):}|

|{:(-1,b,c),(a,-1,c),(a,b,-1):}|=(a+1)(b+1)(c+1)((a)/(a+1)+(b)/(b+1)+(c)/(c+1)-1)

Without expanding the determinant, prove that |{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|=(a-b)(b-c)(c-a) .

|{:(1,a,bc),(1,b,ca),(1,c,ab):}|=|{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|

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

Let a,b,c be such that b(a+c) ne 0. If |{:(a,a+1,a-1),(-b,b+1,b-1),(c,c-1,c+1):}|+|{:(a+1,b+1,c-1),(a-1,b-1,c+1),((-1)^(n+2)a,(-1)^(n+1)b,(-1)^(n)c):}| =0 then the value of n is

CHHAYA PUBLICATION-DETERMINANT -EXERCISE 2B
  1. |{:(a,b,c),(x,y,z),(p,q,r):}|=|{:(x,y,z),(p,q,r),(a,b,c):}|=|{:(y,b,q)...

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  2. |{:(1,omega,omega^2),(omega,omega^2,1),(omega^2,1,omega):}|=0 where om...

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  3. |{:(b+c,a,1),(c+a,b,1),(a+b,c,1):}|=0

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  4. |{:(a-b,1,a),(b-c,1,b),(c-a,1,c):}|=|{:(a,1,b),(b,1,c),(c,1,a):}|

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  5. |{:(a+1,a+4,a+2),(a+2,a+5,a+4),(a+3,a+6,a+6):}|=0

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  6. |{:(" "0," "a,b),(-a," "0,c),(-b,-c,0):}|

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  7. |{:(9,9,12),(1,-3,-4),(1,9,12):}|=0

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  8. |{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}|=|{:(1,a^2,a^3),(1,b^2,b^3),(1,c...

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  9. |{:(441,442,443),(445,446,447),(449,450,451):}|=0

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  10. Prove |{:(cos(x-a),cos(x+a),cosx),(sin(x+a),sin(x-a),sinx),(cosa tan x...

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  11. |{:(101,103,105),(104,105,106),(107,108,109):}|=0

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  12. |{:(5^2,5^3,5^4),(5^3,5^4,5^5),(5^4,5^6,5^7):}|=0

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  13. |{:(1,alpha,alpha^2),(1,beta,beta^2),(1,gamma,gamma^2):}|=(alpha-beta)...

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  14. |{:(1,1,1),(a,b,c),(bc,ca,ab):}|=(a-b)(b-c)(c-a)

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  15. |{:(1,alpha,alpha^3),(1,beta,beta^3),(1,gamma,gamma^3):}|=(alpha-beta)...

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  16. |{:(a+b+2c," "a," "b),(" "c,b+c+2a," "b),(" "c,...

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  17. |{:(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=(b-c)(c-a)(a-b)(bc+ca+ab)

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  18. |{:(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=abc(a-b)(b-c)(c-a)

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  19. |{:(a,b,c),(a^2,b^2,c^2),(b+c,c+a,a+b):}|=(a-b)(b-c)(c-a)(a+b+c)

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  20. |{:(1+x,1,1),(1,1+y,1),(1,1,1+z):}|=xy+yz+zx+xyz

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