|{:(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c):}...

`|{:(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c):}|=3abc-a^3-b^3-c^3`

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|{:(b+c,a-c,a-b),(b-c,c+a,b-a),(c-b,c-a,a+b):}|=8abc

|{:(a,b,c),(a-b,b-c,c-a),(b+c,c+a,a+b):}|=a^3+b^3+c^3-3abc

Prove |{:(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ab,c^3):}|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)

|{:(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=abc(a-b)(b-c)(c-a)

Using properties of determinants , prove that, |{:(a,b,c),(b,c,a),(c,a,b):}|=-(a^3+b^3+c^3-3abc) and hence show that, |{:(2bc-a^2," "c^2," "b^2),(" "c^2,2ca-b^2," "a^2),(" "b^2," "a^2,2ab-c^2):}|=(a^3+b^3+c^3-3abc)^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)

Prove that, |(2a, a-b-c,2a),(2b,2b,b-c-a),(c-a-b,2c,2c)|=(a + b+c)^3

Show that abs((a,b,c),(a-b,b-c,c-a),(b+c,c+a,a+b)) = a^3+b^3+c^3 -3abc.

|{:(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=(b-c)(c-a)(a-b)(bc+ca+ab)

{:|( 3a,-a+b,-a+c),( -b+a, 3b,-b+c) ,( -c+a,-c+b,3c) |:} = 3( a+b+c) ( ab+bc+ca)

CHHAYA PUBLICATION-DETERMINANT -EXERCISE 2B

|{:(" "3a," "-a+b,-a+c),(a-b," "3b," "c-b),(a-c," "b-c," "3c):}|...
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|{:(x,a,b),(a,x,b),(a,b,x):}|=(x-a)(x-b)(x+a+b)
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|{:(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c):}|=3abc-a^3-b^3-c^3
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Prove |{:(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ab,c^3):}|=-(a-b)(b-c)(c...
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|{:(a^2,a^2-(b-c)^2,bc),(b^2,b^2-(c-a)^2,ca),(c^2,c^2-(a-b)^2,ab):}|=(...
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|{:((a+1)(a+2),a+2,1),((a+2)(a+3),a+3,1),((a+3)(a+4),a+4,1):}|=-2
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Show that the following determinant is a perfact square : |{:(1,a,a^...
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if x+y+z = 0 then prove that |{:(x,y,z),(x^2,y^2,z^2),(y+z,z+x,x+y):}|...
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|{:(1,1,1),(x,y,z),(x^3,y^3,z^3):}|=0
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|{:(x," "4,-2),(4," "x,-2),(4,-2," "x):}|=0
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|{:(x,a,a),(a,x,b),(b,b,x):}|=0
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|{:(x+1," "2," "3),(" "1,x+1," "3),(" "3,-6,x+1):}|=0
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|{:(a-x," "b," "c),(" "b,c-x," "a),(" "c," "a,b-x):}|=0
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|{:(" "x," "c+x," "b+x),(" "c+x," "x," "a+x),(" "b+x," "a+x," ...
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|{:(2-x," "2," "3),(" "2,5-x," "6),(" "3," "4,10-x):}|=0
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solve:|{:(3+x,3-x,3-x),(3-x,3+x,3-x),(3-x,3-x,3+x):}|=0
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|{:(x-2,2x-3,3x-4),(x-4,2x-9,3x-16),(x-8,2x-27,3x-64):}|=0
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|{:(0,x-a,x-b),(x+a,0,x-c),(x+b,x+c,0):}|=0[where" "a neb nec]
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Solve: |{:(3,5,x),(7,x,7),(x,5,3):}|=0
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|{:(x+a," "b," "c),(" "a,x+b," "c),(" "a," "b,x+c):}|=0
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