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suppose D= |{:(a(1),,b(1),,c(1)),(a(2)...

suppose `D= |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}| ` and
`D^(r )= |{:(a_(1)+pb_(1),,b_(1)+qc_(1),,c_(1)+ra_(1)),(a_(2)+pb_(2),,b_(2)+qc_(2),,c_(2)+ra_(2)),(a_(3)+pb_(3),,b_(3)+qc_(3),,c_(3)+ra_(3)):}| `. Then

A

`D' = D`

B

`D' = D (1 - pqr)`

C

`D' = D ( 1 + p + q + r)`

D

`D' = D (1 + pqr)`

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The correct Answer is:
A
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suppose D= |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}| and Dprime= |{:(a_(1)+pb_(1),,b_(1)+qc_(1),,c_(1)+ra_(1)),(a_(2)+pb_(2),,b_(2)+qc_(2),,c_(2)+ra_(2)),(a_(3)+pb_(3),,b_(3)+qc_(3),,c_(3)+ra_(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))|

If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =5 , then the value of Delta = |(b_(2) c_(3) - b_(3) c_(2),a_(3) c_(2) - a_(2) c_(3),a_(2) b_(3) -a_(3) b_(2)),(b_(3) c_(1) - b_(1) c_(3),a_(1) c_(3) - a_(3) c_(1),a_(3) b_(1) - a_(1) b_(3)),(b_(1) c_(2) - b_(2) c_(1),a_(2) c_(1) - a_(1) c_(2),a_(1) b_(2) - a_(2) b_(1))| is

If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =5 , then the value of Delta = |(b_(2) c_(3) - b_(3) c_(2),a_(3) c_(2) - a_(2) c_(3),a_(2) b_(3) -a_(3) b_(2)),(b_(3) c_(1) - b_(1) c_(3),a_(1) c_(3) - a_(3) c_(1),a_(3) b_(1) - a_(1) b_(3)),(b_(1) c_(2) - b_(2) c_(1),a_(2) c_(1) - a_(1) c_(2),a_(1) b_(2) - a_(2) b_(1))| is

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(2),c_(2)),(b_(3),c_(3)):}|-b_(1)|{:(a_(2),c_(2)),(a_(3),c_(3)):}|+c_(1)|{:(a_(2),b_(2)),(a_(3),b_(3)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column The value of k for which determinant |{:(2,3,-1),(-1,-2,k),(1,-4,1):}| vanishes, is "(a) -3 (b) 7/11 (c) -2 (d) 2"

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(2),c_(2)),(b_(3),c_(3)):}|-b_(1)|{:(a_(2),c_(2)),(a_(3),c_(3)):}|+c_(1)|{:(a_(2),b_(2)),(a_(3),b_(3)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column The value of the determinant |{:(2,3,4),(6,5,7),(1,-3,2):}|is: "(a) 54 (b) 40 (c) -45 (d) -40"

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(2),c_(2)),(b_(3),c_(3)):}|-b_(1)|{:(a_(2),c_(2)),(a_(3),c_(3)):}|+c_(1)|{:(a_(2),b_(2)),(a_(3),b_(3)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column. The vaue of the determinant |{:(5,1),(3,2):}|is: "(a) 4 (b) 5 (c) 6 (d) 7 "

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 a_(1),a_(2),a_(3)……,a_(12) are in A.P and Delta_(1)= |{:(a_(1)a_(5),,a_(1),,a_(2)),(a_(2)a_(6),,a_(2),,a_(3)),(a_(3)a_(7),,a_(3),,a_(4)):}| Delta_(2)= |{:(a_(2)a_(10),,a_(2),,a_(3)),(a_(3)a_(11),,a_(3),,a_(4)),(a_(4)a_(12),,a_(4),,a_(5)):}| then Delta_(1):Delta_(2)= "_____"

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

OBJECTIVE RD SHARMA ENGLISH-DETERMINANTS-Exercise
  1. Let omega=-1/2+i(sqrt(3))/2dot Then the value of the determinant |1 1 ...

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  2. If a+b+c=0, one root of |a-x c b c b-x a b a c-x|=0 is x=1 b. x=2 c. ...

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  3. suppose D= |{:(a(1),,b(1),,c(1)),(a(2),,b(2),,c(2)),(a(3),,b(3),,c(3...

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  4. A and B are two non-zero square matrices such that AB = O. Then,

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  5. The roots of the equation |{:(x-1,1,1),(1,x-1,1),(1,1,x-1):}|=0 are

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  6. From the matrix equation AB=AC, we conclude B=C provided.

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  7. If A=|[1, 1, 1],[a, b, c],[ a^2,b^2,c^2]| , B=|[1,bc, a],[1,ca, b],[1,...

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  8. The value of |(11,12,13),(12,13,14),(13,14,15)|, is

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  9. |{:(x,4, y+z),(y,4,z+x),(z,4,x+y):}| is equal to:

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

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  11. Let a ,b , c be the real numbers. The following system of equations in...

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  12. A ,\ B are two matrices such that A B and A+B are both defined; sho...

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  13. If omega is an imaginary cube root of unity, then the value of |(a,b o...

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  14. If alpha, beta are non - real numbers satifying x^3-1=0 then the value...

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  15. The value of the determinant |(-1,1,1),(1,-1,1),(1,1,-1)| is equal to

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  16. In a third order determinant, each element of the first column consist...

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  17. A root of the equation |[3-x,-6,3],[-6,3-x,3],[3,3,-6-x]|=0

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  18. For positive numbers x, y and z, the numerical value of the determinan...

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  19. Calculate the value of the determinant |{:(1,1,1,1),(1,2,3,4),(1,3,6,1...

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  20. if Delta= |{:(3,,4,,5,,x),(4,,5,,6,,y),(5,,6,,7,,z),(x,,y,,z,,0):}|=0 ...

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