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Prove the identities: |[b^2+c^2,ab, ac]...

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`

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$$ \left|\begin{array}{ccc} b^{2}+c^{2} & a b & a c \\ a b & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2} \end{array}\right| $$ by Multiplying `R_{1}, R_{2}, R_{3}` with `a b, c ` we get ...
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Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|=4a^2b^2c^2

|[b^2c^2,bc,b+c] , [c^2a^2,ca,c+a] , [a^2b^2,ab,a+b]|=0

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 |[-a^(2),ab,ac],[ba,-b^(2),bc],[ca,cb,-c^(2)]|=4a^(2)b^(2)c^(2)

Prove that identities: |[-bc,b^2+bc,c^2+bc],[a^2+ac,-ac,c^2+ac],[a^2+ab,b^2+ab,-ab]|=(a b+b c+a c)^3

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.

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

If A=[[a^2,ab,ac],[ab,b^2,bc],[ac,bc,c^2]] and a^2+b^2+c^2=1, then A^2

Given : a^(2)+b^(2)+c^(2) =0 Prove the following : |{:(b^(2)+c^(2),ab,ca),(ab,c^(2)+a^(2),bc),(ca,bc,a^(2)+b^(2)):}|=4a^(2)b^(2)c^(2)

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]|

RD SHARMA-DETERMINANTS-Solved Examples And Exercises
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  2. Prove the identities: |[a, b-c,c-b],[ a-c, b, c-a],[ a-b,b-a, c]| =(...

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  3. Prove the identities: |[b^2+c^2,ab, ac],[ba,c^2+a^2,bc],[ca, cb ,a^2+...

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  4. |((a^2+b^2)/c,c,c),(a,(b^2+c^2)/a,a),(b,b,(a^2+c^2)/b)| equal to : (A)...

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  5. Prove that identities: |[-bc,b^2+bc,c^2+bc],[a^2+ac,-ac,c^2+ac],[a^2+...

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  6. 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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  7. If |[p, b, c],[ a, q, c],[ a, b, r]|=0, find the value of p/(p-a)+q/(...

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  10. If f(x)=|0x-a x-b x+a0x-c x+b x+c0|,t h e n f(x)=0 (b) f(b)=0 (c) f...

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  11. Prove the identities: |[a^2,a^2-(b-c)^2,b c], [b^2,b^2-(c-a)^2,c a],[...

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  12. Prove the identities: |[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x,...

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  13. Without expanding, show that the value of each of the determinants is ...

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  14. Without expanding, show that the value of the determinant is zero: ...

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  15. Without expanding, show that the value of each of the determinants is ...

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  16. Without expanding, show that the value of each of the determinants is ...

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  17. Using the properties of determinants, prove that following |(a-b,-c^2,...

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  18. Prove the identities: |[a, b, c],[ a-b,b-c,c-a],[ b+c,c+a, a+b]|=a^3+...

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  19. For any triangleABC, the value of determinant |[sin^2A,cotA,1],[sin^2B...

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  20. Without expanding, show that the value of each of the determinants is ...

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