Evaluate |{:(.^(x)C(1),,.^(x)C(2),,.^(x)...

Evaluate `|{:(.^(x)C_(1),,.^(x)C_(2),,.^(x)C_(3)),(.^(y)C_(1),,.^(y)C_(2),,.^(y)C_(3)),(.^(x)C_(1),,.^(z)C_(2),,.^(z)C_(3)):}|`

xyz (x-y)(y-z)(z-x)

`(xyz)/(6) (x-y)(y-z)(z-x)`

`(xyz)/(12) (x-y)(y-z)(z-x)`

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|{:(.^(x)C_(r),,.^(x)C_(r+1),,.^(x)C_(r+2)),(.^(y)C_(r),,.^(y)C_(r+1),,.^(y)C_(r+2)),(.^(z)C_(r),,.^(z)C_(r+1),,.^(z)C_(r+2)):}| is equal to

The determinant |^(^^)xC_(1),^(x)C_(2),^(x)C_(3)^(^^)yC_(1),^(y)C_(2),^(y)C_(3)sim zC_(1),zC_(2),zC_(3)] is equal to-

Evaluate det[[x_(C_(1)),x_(C_(2)),x_(C_(3))y_(C_(1)),y_(C_(2)),yC_(3)z_(C_(1)),z_(C_(2)),z_(C_(3))]]

prove that |{:((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)):}| |{:((1+ax)^(2),,(1+bx)^(2),,(1+cx)^(2)),((1+ay)^(2),,(1+by)^(2),,(1+cy)^(2)),((1+az)^(2),,(1+bx)^(2),,(1+cz)^(2)):}| =2 (b-c)(c-a)(a-b)xx (y-z) (z-x)(x-y)

if x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|

STATEMENT-1: If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are collinear, then |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 STATEMENT-2: If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) will be collinear. STATEMENT-3: If lines a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0 are concurrent then |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0

Show that |^x C_r^x C_(r+1)^x C_(r+2)^y C_r^y C_(r+1)^y C_(r+2)^z C_r^z C_(r+1)^z C_(r+1)|=|^x C_r^(x+1)C_(r+1)^(x+2)C_(r+2)^y C_r^(y+1)C_(r+1)^(y+2)C_(r+2)^z C_r^(z+1)C_(r+1)^(z+2)C_(r+1)| .

Prove that the value of each the following determinants is zero: (a^(x)+a^(-x))^(2),(a^(x)-a^(-x))^(2),1(b^(y)+b^(-y))^(2),(b^(y)-+b^(-y))^(2),1(c^(z)+c^(-z))^(2),(c^(z)-c^(-z))^(2),1]|

DISHA PUBLICATION-DETERMINANTS-EXERCISE -2 CONCEPT APPLICATOR

If omega is a non-real cube root of unity, then Delta = |(a(1) + b(1) ...
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prove that |{:((a-x)^(2),,(a-y)^(2),,(a-z)^(2)),((b-x)^(2),,(b-y)^(2),...
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Evaluate |{:(.^(x)C(1),,.^(x)C(2),,.^(x)C(3)),(.^(y)C(1),,.^(y)C(2),,....
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the value of the determinant |{:((a(1)-b(1))^(2),,(a(1)-b(2))^(2),,...
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f'(x) = |{:(cos x , x , 1),(2sin x,x^(2),2x),(tan x, x,1):}| . The val...
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If the matrix A = [{:(y+a,b,c),(a,y+b,c),(a,b,y+c):}] has rank 3, then
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If the system of linear equations x1 + 2x2 + 3x3 = 6, x1 + 3x2 + 5x3 =...
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Let lambda "and" alpha be real. Let S denote the set of all values of ...
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For what value of p, is the system of equation p^3x+(p+1)^3y=(p+2)^3 a...
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