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If x+y+z=0 , prove that |[xa, yb,zc],[y...

If `x+y+z=0` , prove that `|[xa, yb,zc],[yc, za, xb], [zb, xc, ya]|=x y z|[a, b, c],[c ,a, b],[b ,c,c]|`

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To prove that if \( x + y + z = 0 \), then \[ \begin{vmatrix} xa & yb & zc \\ yc & za & xb \\ zb & xc & ya \end{vmatrix} ...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. If a,b,c are all distinct and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-...

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  2. Solve : |(x-2, 2x-3, 3x-4), (x-4, 2x-9, 3x-16) ,(x-8 ,2x-27,3x-64)|=0

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  3. If x+y+z=0 , prove that |[xa, yb,zc],[yc, za, xb], [zb, xc, ya]|=x y ...

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  4. 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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  5. 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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  6. |[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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  7. 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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  8. 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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  9. Prove that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)...

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

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

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  13. 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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  14. 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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  15. If a , b , c , are roots of the equation x^3+p x+q=0, prove that |[a...

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

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

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

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

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