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Show that |1b c a(b+c)1c a b(c+a)1a b c(...

Show that `|1b c a(b+c)1c a b(c+a)1a b c(a+b)|=0` .

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The correct Answer is:
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The given determinant

`[[1 ,bc,a(b+c)],[1,ca, b(c+a)],[1,ab,c(a+b)]]=0`

`[[1 ,bc,(ab+bc+ca)],[1,ca, (bc+ab+ca)],[1,ab,(ca+ab+bc)]]=0`

[applyingC2​→C3​+C2​]

`=(ab+bc+ca)``[[1 ,bc,1],[1,ca,1],[1,ab,1]]=0`

`=(ab+bc+ca)×0 = 0`
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Show that |1a b+c1b c+a1c a+b|=0 .

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  2. Show that |[b-c,c-a, a-b],[ c-a, a-b,b-c],[ a-b,b-c,c-a]| = 0 .

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  3. Show that |1b c a(b+c)1c a b(c+a)1a b c(a+b)|=0 .

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  4. Without expanding prove that: |x+y y+z z+x z x y1 1 1|=0 .

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  5. Without expanding show that: =|cos e c^2thetacot^2theta1cot^2thetac...

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  6. Find the value of the determinant =|2 3 4 5 6 8 6x9x 12 x| .

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  7. Without expanding show that |b^2c^2b c b+cc^2a^2c a c+a a^2b^2a b a...

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  8. Without expanding evaluate the determinant |sinalphacosalphasin(alp...

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  9. Without expanding evaluate the determinant |(x^2+x^(-1))^2(a^2-a^(-1))...

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  10. If a ,\ b ,\ c are in A.P., find the value of |2y+4 5y+7 8y+a3y+5 6...

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  11. Without expanding evaluate the determinant =|265 240 219 240 225 19...

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  12. If 1=|1 1 1x^2y^2z^2x y z| and 2=|1 1 1y z z xx y x y z| , without exp...

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  13. Let =|A xx^2 1B y y^2 1C z z^2 1| and 1=|A B C x y z y z z xx y| , the...

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  14. If 1=|a b c x y z p q r| and 2=|q-b y-p a-x r-c z| , without expanding...

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  15. If A is a skew-symmetric matrix of odd order n , then |A|=O .

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  16. Prove that: |0a-b-a0-c b c0|=0 .

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  17. Without expanding or evaluating show that |0b-a c-a a-b0c-b a-c b-c...

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  18. Without expanding, prove that |a+b x c+dx p+q x a x+b c x+d p x+q u...

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  19. Prove that: |-a^2a b a c b a-b^2b c a c b c-c^2|=4a^2b^2c^2 .

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  20. Prove that: |1 1 1 1 1+x1 1 1 1+y|=x y .

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