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Without expanding the determinant, show that `(a+b+c)` is a factor of the determinant `|a b c b c a c a b|`

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To show that \( (a + b + c) \) is a factor of the determinant \[ \Delta = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} ...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|=4a^2b^2c^2

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  2. Prove that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)...

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  3. Without expanding the determinant, show that (a+b+c) is a factor of th...

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  4. If m in N and mgeq2, prove that: |1 1 1m(C1)m+1(C1)m+2(C1)m(C2)m+1(C2...

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  5. Evaluate: =|10 ! 11 ! 12 ! 11 ! 12 ! 13 ! 12 ! 13 ! 14 !|

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  6. Show that: |b+cc+a a+b q+r r+p p+q y+z z+xx+y|=2|a b c p q r x y z|

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  7. Prove that |(1+a,1, 1),( 1,a+b,1),( 1, 1, 1+c)|=a b c(a+1/a+1/b+1/c)=a...

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

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  9. If a+b+c!=0 and |(a, b, c),(b, c, a),(c,a,b)|=0 , then prove that a=b=...

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

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

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

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  13. In a A B C ,if|(1 ,1 ,1),( 1+cosA,1+cosB,1+cosC),(cos^2A+cosB,cos^2A+...

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  14. In a triangleABC, if |(1,1,1),(1+sinA,1+sinB,1+sinC),(sinA+sin^2A,sinB...

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  15. Show that : |x y z x^2y^2z^2x^3y^3z^3|=x y z(x-y)(y-z)(z-x)dot

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

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

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  18. Show that: |x p q p x q q q x|=(x-p)(x^2+p x-2q^2)dot

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  19. If f(x)=|(a,-1 ,0),(ax,a,-1),(a x^2,a x, a)|, find the value of f(2x)-...

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

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