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Using properties of determinants. Prove ...

Using properties of determinants. Prove that `|3a-a+b-a+c-b+a3b-b+c-c+a-c+b3c|=(1+p x y z)(x-y)(y-z)(z-x3(a+b+c)(a b+b c+c a)`

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$$ \begin{aligned} &L H S=\left|\begin{array}{ccc} 3 a & -a+b & -a+c \\ -b+a & 3 b & -b+c \\ -c+a & -c+b & 3 c \end{array}\right| \\ \\ &=\left|\begin{array}{ccc} ...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. If a , b , c are real numbers, prove that |a b c b c a c a b|=-(a+b+c)...

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  2. Show that: |a b-cc+b a+c b c-a a-bb+a c|=(a+b+c)(a^2+b^2+c^2)dot

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  3. Using properties of determinants. Prove that |3a-a+b-a+c-b+a3b-b+c-c+a...

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  4. Using properties of determinants, solve for x:|a+x a-x a-x a-x a+x a...

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  5. Using properties of determinants, solve the following for x: |x-2 ...

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  6. If a,b,c are all distinct and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-...

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  7. If a , b , c are all positive and are p t h ,q t ha n dr t h terms of ...

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  8. If x+y+z=0 prove that |a x b y c z c y a z b x b z c x a y|=x y z|a b ...

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  9. Prove that: |b c-a^2c a-b^2a b-c^2c a-b^2a b-c^2b c-a^2a b-c^2b c-a^2c...

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  10. f(x)=|(secx,cos x,sec^2x + cot x cosecx),(cos^2x,cos^2x,cosec^2 x),(1,...

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  11. Let "Delta"r=|r x(n(n+1))/2 2r-1y n^2 3r-2z(n(3n-1))/2| . Show that su...

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  12. If "Delta"r=|2^(r-1)2. 3^(r-1)4. 5^(r-1)x y z2^n-1 3^n-1 5^n-1|dot Sho...

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  13. If m is a positive integer and Dr=|(2r-1,\ ^m Cr,1),(m^2-1, 2^m ,m+1...

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  14. Without expanding evaluate the determinant "Delta"=|(1, 1, 1),(a, b,...

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  15. Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y...

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  16. Prove that: |-2a a+b a+c b+a-2bb+cc+a c+b-2c|=4(a+b)(b+c)(c+a)

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  17. Without expanding, show that the value of each of the following det...

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  18. Prove: |(1,b+c ,b^2+c^2),( 1,c+a ,c^2+a^2),( 1,a+b ,a^2+b^2)|=(a-b)(b-...

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  19. Prove: |(a, a+b, a+2b),( a+2b, a ,a+b ),(a+b, a+2b, a)|=9(a+b)b^2

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  20. Prove: |(1,a, b c),(1,b ,c a),(1,c ,a b)|=|(1,a ,a^2),( 1,b,b^2),( 1,c...

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