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Prove: |b+c a a b c+a b cc a+b|=4a b c...

Prove: `|b+c a a b c+a b cc a+b|=4a b c`

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`LHS=Delta=|[b+c,a,a],[b,c+a,b],[c,c,a+b]|`
Applying `underset(1)(R)->underset(1)(R)-underset(2)(R)-underset(3)(R)`
`implies |[b-c-b-c,a-c-a-c,a-b-a-b],[b,c+a,b],[c,c,a+b]|`
`implies |[0,-2c,-2b],[b,c+a,b],[c,c,a+b]|`
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Prove: |a^2+1a b a c a bb^2+1b cc a c b c^2+1|=1+a^2+b^2+c^2

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  2. Prove: |1a a^2a^2 1a a a^2 1|=(a^3-1)^2

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  3. Prove: |b+c a a b c+a b cc a+b|=4a b c

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  4. 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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  5. Prove: |(0,b^2a, c^2a),( a^2b,0,c^2b),( a^2c, b^2c,0)|=2a^3b^3c^3

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  6. Prove: |((a^2+b^2)/c,c,c),( a,(b^2+c^2)/a ,a),( b,b,(c^2+a^2)/b)|=4a b...

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  7. Prove: |(-bc, b^2+b c,c^2+bc), (a^2+a c,-a c,c^2+a c),( a^2+a b,b^2+a ...

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  8. |[x+lambda, 2x, 2x], [2x, x+lambda, 2x], [2x, 2x, x+lambda]| =(5x+ lam...

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  9. सारणिकों के गुणधर्मो का प्रयोग करके प्रश्न 6 से 14 तक को सिद्ध कीजिए: ...

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  10. Prove: |(y+z, z, y),( z, z+x,x),( y, x,x+y)|=4\ x y z

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  11. Show that | (-a(b^2 + c^2 - a^2), 2b^3, 2c^3), (2a^3, -b(c^2 + a...

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  12. Prove: |(1+a,1, 1),( 1, 1+a, 1),(1, 1, 1+a)|=a^3+3a^2

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  13. Prove: |(2y, y-z-x,2y),(2z,2z, z-x-y),( x-y-z,2x,2x)|=(x+y+z)^3

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  14. |[y+z ,x, y],[ z+y, z, x],[x+y, y ,z]|=(x+y+z)(x-z)^2

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  15. Using properties of determinants, prove that |(a+x, y, z),( x, a+y, z...

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  16. Prove: |a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a)(a+b+c)

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  17. Without expanding, prove that |(a, b, c),( x, y, z),( p, q ,r)|=|(x, y...

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  18. Show that |x+1x+2x+a x+2x+3x+b x+3x+4x+c|=0 where a ,\ b ,\ c are in A...

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  19. Show that [[x-3,x-4,x-alpha],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]=0 whe...

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  20. If a ,\ b ,\ c are real numbers such that |b+cc+a a+b c+a a+bb+c a+...

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