|{:(a,b,c),(x,y,z),(p,q,r):}|=|{:(x,y,z)...

`|{:(a,b,c),(x,y,z),(p,q,r):}|=|{:(x,y,z),(p,q,r),(a,b,c):}|=|{:(y,b,q),(x,a,p),(z,c,r):}|`

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Using the property of determinants and without expanding {:|( b+c,q+r,y+z),( c+a,r+p,z+x),( a+b,p+q,x+y) |:}=2 {:|(a,q,x),( b,q,y),(c,r,z)|:}

prove that, |{:(x,b,c),(a,y,c),(a,b,z):}|=(x-a)(y-b)(z-c)((x)/(x-a)+(y)/(y-b)+(z)/(z-c)-2)

Show that abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z)) =0

If A=[(a, b, c), (x, y, z), (p, q, r)], B=[(q,-b, y),(-p, a,-x),(r,-c, z)] and if A is invertible, then which of the following is not true? (a) |A|=|B| (b) |A|=-|B| (c) |adj A|=|adj B| (d) A is invertible if and only if B is invertible

|{:(" "a," "b," "c),(" "a+2x," "b+2y," "c+2z),(" "x," "y," "z):}|

Prove that |{:((b+x)(c+x),,(c+x)(a+x),,(a+x)(b+x)),((b+y)(c+y) ,,(c+y)(a+y),,(a+y)(b+y)),((b+z)(c+z),, (c+z)(a+z),,(a+z)(b+x)):}| = (b-c) (c-a) (a-b) (y-z) (z-x) (x-y)

If p+-a, q+- b, r+-c and abs((p,b,c),(a,q,c),(a,b,r)) = 0 show that p/(p-a)+q/(q-b)+r/(r-c)=2 .

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, c],[c ,a ,b],[b ,c ,a]|

Show that abs(((a-x)^2,(a-y)^2,(a-z)^2) , ((b-x)^2,(b-y)^2,(b-z)^2) , ((c-x)^2,(c-y)^2, (c-z)^2)) = 2(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)

Let a(a != 0) is a fixed real number and (a-x)/(p x) = (a-y)/(qy) = (a-z)/(r z) . If p, q, r are in A.P., show that (1)/(x), (1)/(y), (1)/(z) are in A.P.

CHHAYA PUBLICATION-DETERMINANT -EXERCISE 2B

|{:(x,a,b),(a,x,b),(a,b,x):}|
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|{:(2,3,4),(3,4,5),(4,5,6):}|=0
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|{:(a,b,c),(x,y,z),(p,q,r):}|=|{:(x,y,z),(p,q,r),(a,b,c):}|=|{:(y,b,q)...
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|{:(1,omega,omega^2),(omega,omega^2,1),(omega^2,1,omega):}|=0 where om...
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|{:(b+c,a,1),(c+a,b,1),(a+b,c,1):}|=0
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|{:(a-b,1,a),(b-c,1,b),(c-a,1,c):}|=|{:(a,1,b),(b,1,c),(c,1,a):}|
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|{:(a+1,a+4,a+2),(a+2,a+5,a+4),(a+3,a+6,a+6):}|=0
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|{:(" "0," "a,b),(-a," "0,c),(-b,-c,0):}|
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|{:(9,9,12),(1,-3,-4),(1,9,12):}|=0
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|{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}|=|{:(1,a^2,a^3),(1,b^2,b^3),(1,c...
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|{:(441,442,443),(445,446,447),(449,450,451):}|=0
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Prove |{:(cos(x-a),cos(x+a),cosx),(sin(x+a),sin(x-a),sinx),(cosa tan x...
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|{:(101,103,105),(104,105,106),(107,108,109):}|=0
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|{:(5^2,5^3,5^4),(5^3,5^4,5^5),(5^4,5^6,5^7):}|=0
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|{:(1,alpha,alpha^2),(1,beta,beta^2),(1,gamma,gamma^2):}|=(alpha-beta)...
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|{:(1,1,1),(a,b,c),(bc,ca,ab):}|=(a-b)(b-c)(c-a)
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|{:(1,alpha,alpha^3),(1,beta,beta^3),(1,gamma,gamma^3):}|=(alpha-beta)...
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|{:(a+b+2c," "a," "b),(" "c,b+c+2a," "b),(" "c,...
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|{:(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=(b-c)(c-a)(a-b)(bc+ca+ab)
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|{:(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=abc(a-b)(b-c)(c-a)
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