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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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To show that \[ \begin{vmatrix} b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2 \end{vmatrix} ...
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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.

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

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=

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

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. |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|...

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  2. Prove that : |[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]|...

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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 that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)...

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  5. Without expanding the determinant, show that (a+b+c) is a factor of th...

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  6. If m in N and mgeq2, prove that: |1 1 1m(C1)m+1(C1)m+2(C1)m(C2)m+1(C2...

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  7. Evaluate: =|10 ! 11 ! 12 ! 11 ! 12 ! 13 ! 12 ! 13 ! 14 !|

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  8. Show that: |b+cc+a a+b q+r r+p p+q y+z z+xx+y|=2|a b c p q r x y z|

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  9. Prove that |(1+a,1, 1),( 1,a+b,1),( 1, 1, 1+c)|=a b c(a+1/a+1/b+1/c)=a...

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  10. If a , b , c , are roots of the equation x^3+p x+q=0, prove that |[a...

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  11. If a+b+c!=0 and |(a, b, c),(b, c, a),(c,a,b)|=0 , then prove that a=b=...

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  12. Let a , band c detnote the sides BC,CA andAB respectively of triangl...

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  13. Prove that |[a^2+2a,2a+1,1],[2a+1,a+2,1],[3,3,1]|=(a-1)^3

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  14. Using properties of determinant show that: |[1 , a , -bc] , [1 , b , -...

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  15. In a A B C ,if|(1 ,1 ,1),( 1+cosA,1+cosB,1+cosC),(cos^2A+cosB,cos^2A+...

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  16. In a triangleABC, if |(1,1,1),(1+sinA,1+sinB,1+sinC),(sinA+sin^2A,sinB...

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  17. Show that : |x y z x^2y^2z^2x^3y^3z^3|=x y z(x-y)(y-z)(z-x)dot

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  18. Without expanding or evaluating show that |[0 , b-a , c-a] , [a-b , 0...

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  19. If A is a skew-symmetric matrix of odd order n , then |A|=0

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  20. Show that: |x p q p x q q q x|=(x-p)(x^2+p x-2q^2)dot

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