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Using properties of determinants, prove ...

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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Answer any three questions Using properties of determinants, prove the following abs{:(1+a^2 - b^2,2ab,-2b),(2ab,1-a^(2) +b^(2) ,2a),(2b,-2a,1-a^2 -b^2):}=(1+a^2 +b^2)^3.

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

Prove the following: [[b^2-ab,b-c,bc-ac],[ab-a^2,a-b,b^2-ab],[bc-ac,c-a,ab-a^2]]=0

Prove the following: [[-a^2,ab,ac],[ab,-b^2,bc],[ac,bc,-c^2]]=4a^2b^2c^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+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]] =1+a^2+b^2+c^2

Using the properties of determinants, show that abs[[1+a^2-b^2,2ab,-2b],[2ab,1-a^2+b^2,2a],[2b,-2a,1-a^2-b^2]]=(1+a^2+b^2)^3

Without expanding the determinants prove that |{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}| = |{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}|

ARIHANT PUBLICATION-DETERMINANTS -CHAPTER PRACTICE
  1. Using properties of determinants, prove the following abs{:(a^2, bc, a...

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  2. If [(3x,7),(-2,4):}]=[(8,7),(6,4):}]then find the value of x.

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  3. IF A=[(1,2),(4,2):}] then show that |2A|=4 |A|

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  4. If [{:(x+1,x-1),(x-3,x+2):}]=[{:(4,-1),(1,3):}]then write the value of...

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  5. If the determinant of matrix A of order 3 times 3 is of value 4, then ...

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  6. If f(x)|{:(0,x-a,x-b),(x+a,0,x-c),(x+b,x+c,0):}| then show that f(0)=0

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  7. Show that without expanding at any stage |{:(1/a,a^2,bc),(1/b,b^2,ca...

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  8. Find the area of triangle , whose vertices are (2,7) (1,1) and (10,8)

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  9. Show that the points (a+5,a-4),(a-2,a+3) and (a,a) do not lie on a st...

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  10. IF Ay is the cofactor of the element ay of the determinant |{:(2,-3,5)...

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  11. For what value of matrix [{:(6-x,4),(3-x,1):}] is a singular matrix?

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  12. For what value of x, A=[(2(x+1),2x),(x,x-2):}]singular matrix

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  13. Find x if [(1,2,x),(1,1,1),(2,1,-1):}]is singular

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  14. IF the value of the third order determinant is 12, then find the value...

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  15. IF for the non singular matrix A,A^2=I then find A^-1

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  16. A is a non singular symmetric matrix, write whether A^-1 is symmetric ...

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  17. 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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  18. Prove that |[x, y, z],[x^2, y^2, z^2], [x^3, y^3, z^3]|=xyz(x-y)(y-z)(...

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  19. Prove the following : [[x+4,2x,2x],[2x,x+4,2x],[2x,2x,x+4]]-(5x+4)(4-x...

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  20. Prove that: |[1, 1+p, 1+p+q],[2, 3+2p, 1+3p+2p], [3, 6+3p, 1+6p+3q ]|=...

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  21. Without expanding the determinants prove that |{:(a,a^2,bc),(b,b^2,c...

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