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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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`|(a,b,c),(b,c,a),(c,a,b)| = /_ `
`C_1 -> C_1 + C_2 + C_3`
`|(a+b+c, b,c),(b+c+a,c,a),(c+a+b,a,b)|`
`(a+b+c)|(1,b,c),(1,c,a),(1,a,b)|= /_`
`(a+b+c)` is a factor of given determinant
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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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  7. 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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  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=cdot

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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 , ca...

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

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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^2,y^2,z^2],[x^3,y^3,z^3]|=x y z(x-y)(y-z)(z...

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

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

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

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