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If `a , b , c ,` are roots of the equation `x^3+p x+q=0,` prove that `|[a ,b, c],[ b ,c, a],[ c, a, b]|=0`

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RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Show that: |b+c c+a a+b q+r r+p p+q y+z z+x x+y|=2|a b c p q r x y z|

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  2. Prove that |[1+a,1, 1], [1,1+b,1], [1, 1, 1+c]|=a b c(1+1/a+1/b+1/c)=a...

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  3. If a , b , c , are roots of the equation x^3+p x+q=0, prove that |[a...

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  4. If a+b+c!=0 and |a b c b c a c a b|=0 , then prove that a=b=cdot

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  5. Let a , band c detnote the sides BC,CA andAB respectively of triangl...

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  6. Prove that |[a^2+2a,2a+1,1],[2a+1,a+2,1],[3,3,1]|=(a-1)^3

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  7. Using properties of determinant show that: |[1 , a , bc] , [1 , b , ca...

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  8. Using properties of determinants, show that triangle ABC is isosceles,...

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  9. In a triangleABC, if |[1,1,1][1+sinA,1+sinB,1+sinC],[sinA+sin^2A, sinB...

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  10. Show that : |[x, y, z ],[x^2,y^2,z^2],[x^3,y^3,z^3]|=x y z(x-y)(y-z)(z...

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  11. Without expanding or evaluating show that |[0 , b-a , c-a] , [a-b , 0...

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  12. If A is a skew-symmetric matrix of odd order n , then |A|=0

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  13. Using properties of determinants, show that |{:(x, p, q), ( p, x, q)...

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  14. If f(x)=|[a,-1 ,0],[ax,a,-1],[a x^2,a x, a]|, using properties of det...

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  15. If a , b , c are distinct real numbers and the system of equations a x...

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  16. If x , y , z are not all zero such that a x+y+z=0, x+b y+z=0, ...

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  17. Find the value of lambda for which the homogeneous system of equations...

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  18. If the system of equations x=c y+b z y=a z+c x z=b x+a y has...

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  19. A matrix A of order 3x3 has determinant 5. What is the value of |3A|?

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  20. If A is a square matrix such that |A|=2 , write the value of |AA^T|

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