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If a, b and c are all positive real, the...

If a, b and c are all positive real, then prove that minimum value of determinant
`|{:(a^2+1,ab,ac),(ab,b^2+1,bc),(ac,bc,c^2+1):}|` = `1+a^2+b^2+c^2`

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Prove the following: [[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]] =1+a^2+b^2+c^2

Find the minimum value of abs[[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]]

Prove the following: [[-a^2,ab,ac],[ab,-b^2,bc],[ac,bc,-c^2]]=4a^2b^2c^2

Show that without expanding at any stage |{:(1/a,a^2,bc),(1/b,b^2,ca),(1/c,c^2,ab):}|=0

Using properties of determinants , prove that |{:(1,a,bc),(1,b,ca),(1,c,ab):}|=(a-b)(b-c)(c-a)

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Using properties of determinants, prove the following abs{:(a^2, bc, ac +c^2 ),(a^(2) + ab, b^(2),ac ),(ab, b^(2) + bc,c^(2) ):}=4a^(2) b^(2) c^(2) .

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USHA PUBLICATION-MATRICES DETERMINANTS PROBABILITY -EXERCISE
  1. Without expanding,show that the following determinant vanishes. abs((6...

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  2. Show that: abs((1,a,a^2),(1,b,b^2),(1,c,c^2))=(a-b)(b-c)(c-a)

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  3. If a, b and c are all positive real, then prove that minimum value of ...

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  4. Show that: abs((a,a^2,a^3),(b,b^2,b^3),(c,c^2,c^3))=abc(a-b)(b-c)(c-a)

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  5. Prove that: |[a, b, c],[a-b, b-c, c-a ], [b+c, c+a, a+b]|= a^3+b^3+c^3...

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  6. Solve by cramer's rule :2x+3y=6, 4x+6y=12

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  7. If A=[[ 3,-5,],[-4,2,]],show that A^2-5A-14I=O,where I is a unit matri...

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  8. Find the matrix B,if B^2=[[16,0,],[0,16,]].

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  9. If A=[[1,2,],[3,4,]],B=[[-2,3,],[-1,2,]],show that (A+B)^T=A^T+B^T.

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  10. If A=[[2,0,3,],[3,4,5,]],B=[[3,-1,2,],[0,2,3,]]and c=[[2,-3,0,],[1,4,5...

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  11. If A=[[1,2,],[3,4,]],B=[[-2,3,],[-1,2,]],show that (A+B)^T=A^T+B^T.

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  12. If A=[[2,0,3,],[3,4,5,]],B=[[3,-1,2,],[0,2,3,]]and c=[[2,-3,0,],[1,4,5...

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  13. Find the matrix A if A^2=[[17,8,],[8,17,]].

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  14. Verify that (AB)^-1=B^-1A^-1 if A=[[2,3,],[1,-1,]],B=[[0,1,],[3,1,]]

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  15. Verify that (AB)^-1=B^-1A^-1 if A=[[1,2,3,],[4,5,6,],[7,8,9,]],B=[[1,2...

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  16. If A=[[1,-1,1,],[2,-1,0,],[1,0,0,]],then show that A^2=A^-1.

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  17. If A=[{:(3,1),(-1,2):}] then prove that A^(2)-5A+7I=O

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  18. Show that the matrix A=[[2,3,],[1,2,]]satisfies the equation A^2-4A+I=...

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  19. If A=[{:(costheta,sintheta),(-sintheta,costheta):}], then prove that ...

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  20. Express the matrix A=[[4,2,-3],[1,3,-6],[-5,0,-7]] as the sum of symme...

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