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Show that: |[a,b-c,c+b],[a+c,b,c-a],[a-b...

Show that: `|[a,b-c,c+b],[a+c,b,c-a],[a-b,b+a,c]|=(a+b+c)(a^2+b^2+c^2)`.

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To show that \[ \left| \begin{array}{ccc} a & b - c & c + b \\ a + c & b & c - a \\ a - b & b + a & c \end{array} \right| = (a + b + c)(a^2 + b^2 + c^2), ...
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Show that |[b-c,c-a, a-b],[ c-a, a-b,b-c],[ a-b,b-c,c-a]| = 0 .

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

Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,c^2],[a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a b+b c+c a) .

Show that |[1,a,a^2],[1,b,b^2],[1,c,c^2]|=(a-b)(b-c)(c-a)

Prove that: |[b, c-a^2,c] ,[a-b^2,a b-c^2,c ],[a-b^2,a ,b-c^2b c-a^2a b-c^2b c-a^2c a-b^2]|=|[a, b, c],[ b ,c ,a],[ c, a ,b]|^2 .

Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c)(c+a)

Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c)(c+a)

By using properties of determinants. Show that: |[a^2+1,a b, a c],[ a b,b^2+1,b c],[c a, c b, c^2+1]|=(1+a^2+b^2+c^2)

By using properties of determinants. Show that: (i) |[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|=(a-b)(b-c)(c-a) (ii) |[1, 1, 1],[a, b, c],[ a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a+b+c)

RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Find a quadratic polynomial phi(x) whose zeros are the maximum and mi...

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  2. Let f(x)=|[secx,cosx,sec^2x+cotxcosecx],[cos^2x,cos^2x,cosec^2x],[1,co...

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  3. Show that: |[a,b-c,c+b],[a+c,b,c-a],[a-b,b+a,c]|=(a+b+c)(a^2+b^2+c^2).

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  4. If a , b , c are real numbers, prove that |[a,b,c],[b,c,a],[c,a,b]|=-(...

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  5. Solve: |a+x a-x a-x a-x a+x a-x a-x a-x a+x|=0

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  6. Show that: |[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]|=3(a+b+c)(a b...

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  7. 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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  8. Solve : |[x-2,2x-3,3x], [-4x-4,2x-9,3x-16], [x-8,2x-27,3x-64]| = 0

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  9. If x+y+z=0 , prove that |[xa, yb,zc],[yc, za, xb], [zb, xc, ya]|=x y ...

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  10. If a , b , c are all positive and are pth ,qth and rth terms of a G.P....

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  11. Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-...

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  12. |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|...

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  13. Prove that: |[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]|=2a...

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

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

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  17. 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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  18. Evaluate: =|(10 !, 11 !, 12 !), (11 !, 12 !, 13 !), (12 !, 13 !, 14 !)...

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  19. 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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  20. 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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