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Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,b...

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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`Delta=|[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|`

Multiplying the 3 rows by a,b and c

`implies |[a(b^2+c^2),a^2b,a^2c],[b^2a,b(c^2+a^2),b^2c],[c^2a,c^2b,c(a^2+b^2)]|`

Taking common a,b and c from columns ...
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Prove the identities: |[b^2+c^2,ab, ac],[ba,c^2+a^2,bc],[ca, cb ,a^2+b^2]|=4a^2b^2c^2

Evaluate |[0,c,b] , [c,0,a] , [b,a,0]| hence show that |[0,c,b] , [c,0,a] , [b,a,0]|^2= |[b^2+c^2,ab,ac] , [ab,c^2+a^2,bc] , [ca,bc,a^2+b^2]|=4a^2b^2c^2

Prove that |[b^2+c^2,ab,ac],[ab,c^2+a^2,bc],[ca,cb,a^2+b^2]| is always positive for real a, b and c.

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If |[b^2+c^2,ab,ac] , [ba,c^2+a^2,bc] , [ca,cb,a^2+b^2]|= a square of a determinant Delta of the third order then Delta=

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Show that |[0,c,b] , [c,0,a] , [b,a,0]|^2=|[b^2+c^2, ab, ac] , [ab, c^2+a^2, bc] , [ac, bc, a^2+b^2]|

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

Using properties of determinants, prove that |[a^2, bc, ac+c^2] , [a^2+ab, b^2, ac] , [ab, b^2+bc, c^2]| = 4a^2b^2c^2

RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Prove: |1a a^2a^2 1a a a^2 1|=(a^3-1)^2

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

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

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  5. 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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  6. 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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  7. |[x+lambda, 2x, 2x], [2x, x+lambda, 2x], [2x, 2x, x+lambda]| =(5x+ lam...

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

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

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

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

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

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

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

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

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

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  17. 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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  18. 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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  19. If a ,\ b ,\ c are real numbers such that |b+cc+a a+b c+a a+bb+c a+...

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  20. a!=p , b!=q,c!=r and |(p,b,c),(a,q,c),(a,b,r)|=0 the value of p/(p-...

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