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Prove: |(1,a, a^2),(a^2, 1,a),( a, a^2, ...

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

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RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Prove: |[a^2, 2ab,b^2],[b^2,a^2, 2ab],[2ab,b^2,a^2]|=(a^3+b^3)^2

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  2. Prove: |(a^2+1,a b ,a c), (a b,b^2+1,b c), (c a,cb,c^2+1)|=1+a^2+b^2...

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

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  4. Prove: |a+b+c-c-b-c a+b+c-a-b-a a+b+c|=2(a+b)(b+c)(c+a)

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  5. Prove: |[b+c,a,a],[b,c+a,b],[c,c,a+b]|=4a b c

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

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  8. 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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  9. 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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  10. By using properties of determinants. Show that: (i) |x+4 2x2x2xx+4 2x2...

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  11. show that |{:(y+k,y,y),(y,y+k,y),(y,y,y+k):}|=k^(2)(3y+k)

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

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

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

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

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  16. show that |[y+z ,x, y],[ z+x, z, x],[x+y, y ,z]|=(x+y+z)(x-z)^2

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

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

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

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  20. Show that |[x+1,x+2,x+a], [x+2,x+3,x+b],[ x+3,x+4,x+c]|=0 where a ,\ b...

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