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|[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,b...

`|[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|=|[a,b,c],[b,c,a],[c,a,b]|^2`

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Given,`|[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|`
This matrix is cofactor matrix of
=`​|[a,b,c],[​b,c,a],[​c,a,b]|`
​ Thus, by property of cofactor matrix,
`|[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|=|[a,b,c],[b,c,a],[c,a,b]|^2`
​
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det[[bc-a^(2),ca-b^(2),ab-c^(2)ca-b^(2),ab-c^(2),bc-a^(2)ab-c^(2),bc-a^(2),ca-b^(2)]]=det[[a,b,cb,c,ac,a,b]]^(2)

Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]| is divisible by a+b+c and find the quotient.

Show that |{:(bc-a^(2),,ca-b^(2),,ab-c^(2)),(ca-b^(2),,ab-c^(2),,bc-a^(2)),(ab-c^(2),,bc-a^(2),,ca-b^(2)):}| |{:(a^(2),,c^(2),,2ca-b^(2)),(2ab-c^(2),,b^(2),,a^(2)),(b^(2),,2ac-a^(2),,c^(2)):}|.

Let a, b and c are the roots of the equation x^(3)-7x^(2)+9x-13=0 and A and B are two matrices given by A=[(a,b,c),(b,c,a),(c,a,b)] and B=[(bc-a^(2),ca-b^(2),ab-c^(2)),(ca-b^(2),ab-c^(2),bc-a^(2)),(ab-c^(2),bc-a^(2),ca-b^(2))] , then the value |A||B| is equal to

|[bc,ca,ab],[(b+c)^(2),(c+a)^(2),(a+b)^(2)],[a^(2),b^(2)c^(2)]|

|[b^2c^2,bc,b+c] , [c^2a^2,ca,c+a] , [a^2b^2,ab,a+b]|=0

|[b^(2)c^(2),bc,a-c],[c^(2)a^(2),ca,b-c],[a^(2)b^(2),ab,0]|=?

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Prove |[-bc, b^2+bc, c^2+bc] , [a^2+ac, -ac, c^2+ac] , [a^2+ab, b^2+ab, -ab]|=(ab+bc+ca)^2

RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. If a , b , c are all positive and are p t h ,q t ha n dr t h terms of ...

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  2. Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-...

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  3. |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|...

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  4. Prove that : |[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]|...

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  5. Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|=4a^2b^2c^2

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  6. Prove that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)...

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  7. Without expanding the determinant, show that (a+b+c) is a factor of th...

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  8. If m in N and mgeq2, prove that: |1 1 1m(C1)m+1(C1)m+2(C1)m(C2)m+1(C2...

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  9. Evaluate: =|10 ! 11 ! 12 ! 11 ! 12 ! 13 ! 12 ! 13 ! 14 !|

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  10. Show that: |b+cc+a a+b q+r r+p p+q y+z z+xx+y|=2|a b c p q r x y z|

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  11. Prove that |(1+a,1, 1),( 1,a+b,1),( 1, 1, 1+c)|=a b c(a+1/a+1/b+1/c)=a...

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  12. If a , b , c , are roots of the equation x^3+p x+q=0, prove that |[a...

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  13. If a+b+c!=0 and |(a, b, c),(b, c, a),(c,a,b)|=0 , then prove that a=b=...

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  14. Let a , band c detnote the sides BC,CA andAB respectively of triangl...

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  15. Prove that |[a^2+2a,2a+1,1],[2a+1,a+2,1],[3,3,1]|=(a-1)^3

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  16. Using properties of determinant show that: |[1 , a , -bc] , [1 , b , -...

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  17. In a A B C ,if|(1 ,1 ,1),( 1+cosA,1+cosB,1+cosC),(cos^2A+cosB,cos^2A+...

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  18. In a triangleABC, if |(1,1,1),(1+sinA,1+sinB,1+sinC),(sinA+sin^2A,sinB...

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  19. Show that : |x y z x^2y^2z^2x^3y^3z^3|=x y z(x-y)(y-z)(z-x)dot

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  20. Without expanding or evaluating show that |[0 , b-a , c-a] , [a-b , 0...

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