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Suppose a(1),a(2),a(3) are in A.P. and b...

Suppose `a_(1),a_(2),a_(3)` are in A.P. and `b_(1),b_(2),b_(3)` are in H.P. and let `Delta=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))|` then prove that

A

`Delta` is independent of `a_(1),a_(2),a_(3)`

B

`a_(1)-Delta,a_(2)-2Delta, a_(3)-3Delta` are in A.P

C

`b_(1)+Delta,b_(2)+Delta^(2),b_(3)+Delta"are in H.P"`

D

`Delta "is independent of "b_(1),b_(2),b_(3)`

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A, B, C, D
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Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let /_\=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then

The determinant |(b_(1)+c_(1),c_(1)+a_(1),a_(1)+b_(1)),(b_(2)+c_(2),c_(2)+a_(2),a_(2)+b_(2)),(b_(3)+c_(3),c_(3)+a_(3),a_(3)+b_(3))|

The value of the determinant Delta = |(1 + a_(1) b_(1),1 + a_(1) b_(2),1 + a_(1) b_(3)),(1 + a_(2) b_(1),1 + a_(2) b_(2),1 + a_(2) b_(3)),(1 + a_(3) b_(1) ,1 + a_(3) b_(2),1 + a_(3) b_(3))| , is

if Delta=det[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

det[[2a_(1)b_(1),a_(1)b_(2)+a_(2)b_(1),a_(1)b_(3)+a_(3)b_(1)a_(1)b_(2)+a_(2)b_(1),2a_(2)b_(2),a_(2)b_(3)+a_(3)b_(2)a_(1)b_(3)+a_(3)b_(1),a_(3)b_(2)+a_(2)b_(3),2a_(3)b_(3)]]=

If x in R,a_(i),b_(i),c_(i) in R for i=1,2,3 and |{:(a_(1)+b_(1)x,a_(1)x+b_(1),c_(1)),(a_(2)+b_(2)x,a_(2)x+b_(2),c_(2)),(a_(3)+b_(3)x,a_(3)x+b_(3),c_(3)):}|=0 , then which of the following may be true ?

if quad /_=[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

Show that |[a_(1),b_(1),-c_(1)],[-a_(2),-b_(2),c_(2)],[a_(3),b_(3),-c_(3)]|=|[a_(1),b_(1),c_(1)],[a_(2),b_(2),c_(2)],[a_(3),b_(3),c_(3)]|

RESONANCE-MATRICES & DETERMINANT-PART-III

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  2. Which one of the following is wrong?

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  3. Which of the following is true for matrox A=[{:(,1,-1),(,2,3):}]

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  4. Suppose a(1),a(2),a(3) are in A.P. and b(1),b(2),b(3) are in H.P. and ...

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  5. Let theta=(pi)/(5),X=[{:(,cos theta,-sin theta),(,sin theta,cos theta)...

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  6. If Delta=|{:(,x,2y-z,-z),(,y,2x-z,-z),(,y,2y-z,2x-2y-z):}|,then

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  7. Let a.b ,gt 0 and Delta=|{:(,-x,a,b),(,b,-x,a),(,a,b,-x):}|, then

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  8. The determination Delta=|{:(,b,c,balpha+c),(,c,d,calpha+d),(,balpha+c,...

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  9. The determinant Delta=|{:(,a^(2)(1+x),ab,ac),(,ab,b^(2)(1+x),(bc)),(,a...

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  10. If a non-singular matrix and A^(T) denotes the tranpose of A, then

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  11. Let "Let"(x)=|{:(,2sinx,sin^(2)x,0),(,1,2sin x,sin^(2)x),(,0,1,2sin x)...

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  12. Let Delta=|{:(,1,x,x^(2)),(,x^(2),1,x),(,x,x^(2),1):}|,then

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  13. Let f(x)=|{:(,1//x,logx,x^(n)),(,1,-1//n,(-1)^(n)),(,1,a,a^(2)):}| whe...

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  14. If D is determinant of order three of Delta is a determinant formed by...

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  15. Let A,B,C,D be real matrices such that A^(T)=BCD,B^(T)=CDA,C^(T)=DAB a...

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  16. Let A and B be two 2 xx 2 matrix with real entries, If AB=0 and such t...

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  17. If A^(-1)=[{:(,1,-1,0),(,0,-2,1),(,0,0,-1):}] then

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  18. IF A and B are squre matrices of order 3, then the true statement is/a...

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  19. Let M be a 3xx3 non-singular matrix with det(M)=4,"If" M^(-1)"adj(adjM...

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  20. Let AX=B where A is 3 xx 3 and X and B are 3 xx 1 matrices then which ...

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