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If A is a square matrix of order n such ...

If A is a square matrix of order n such that its elements are polynomila in x and its r-rows become identical for x = k, then

A

`(x -k)^(r)` is a factor of `|A|`

B

`(x -k)^(r) -1` is a factor of `|A|`

C

`(x -k)^(r) +1` is a factor `|A|`

D

`(x -k)^(r)` is a factor of A

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The correct Answer is:
A
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OBJECTIVE RD SHARMA-DETERMINANTS-Exercise
  1. If xneynez" and " |{:(x,x^(2),1+x^(3)),(y,y^(2),1+y^(3)),(z,z^(2),1+z^...

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  2. If |(b +c,c +a,a +b),(a +b,b +c,c +a),(c +a,a +b,b +c)| = k |(a,b,c),(...

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  3. If A is a square matrix of order n such that its elements are polynomi...

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  4. If a^(2) + b^(2) + c^(2) = 0 and |(b^(2) + c^(2) ,ab,ac),(ab,c^(2) + a...

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  5. If a^(-1)+b^(-1)+c^(-1)=0 such that |{:(1+a," "1," "1),(" "1,1+b," "1)...

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  6. If alpha,beta "and" gamma are real number without expanding at any sta...

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  7. If A, B and C denote the angles of a triangle, then Delta = |(-1,cos...

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  8. If |(alpha,x,x,x),(x,beta,x,x),(x,x,gamma,x),(x,x,x,delta)| = f(x) - x...

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  9. If x, y , z are in A.P., then the value of the det (A) is , where A ...

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  10. If a != b != c, are value of x which satisfies the equation |(0,x -a...

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  11. If a ,b ,c are different, then the value of |0x^2-a x^3-b x^2+a0x^2+c ...

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  12. The system of equation kx+y+z=1, x +ky+z=k and x+y+kz=k^(2) has no sol...

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  13. Show that: |b^2+c^2a b a c b a c^2+a^2b cc a c b a^2+b^2|=4a^2b^2c^2

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  14. |[2a1b1, a1b2+a2b1, a1b3+a3b1] , [a1b2+a2b1, 2a2b2, a2b3+a3b2] , [a1b3...

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  15. If B is a non-singular matrix and A is a square matrix, then det (B^(-...

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  16. If 0 lt theta lt pi and the system of equations (sin theta) x + y + ...

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  17. If the determinant |(b -c,c -a,a -b),(b' -c',c' -a',a' -b'),(b'' -c'',...

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  18. If a !=b, then the system of equation ax + by + bz = 0 bx + ay + bz ...

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  19. If |(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))|= (a -b) (b -c) (c -a) (a + b+...

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  20. Show that a x+b y+r=0,b y+c z+p=0a n dc z+a x+q=0 are perpendicular to...

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