Using properties of determinants prove t...

Using properties of determinants prove that ((1,1, 1+3x),(1+3y,1,1),(1, 1+3z,1))=9(3xyz+xy+yz+zx)

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1,1,1+3x1+3y,1,11,1+3z,1]|=9(3xyz+xy+yz+zx)

Using properties of determinants, prove that : |{:((x+y)^(2),zx,xy),(zx,(z+y)^(2),xy),(zy,xy,(z+x)^(2)):}|=2xyz(x+y+z)^(3) .

Using properties of determinants, prove the following: |[x,x^2,1+px^3],[y,y^2,1+py^3],[z,z^2,1+pz^3]|=(1+pxyz)(x-y)(y-z)(z-x)

1+x,1,11,1+y,11,1,1+z]|=xy+yz+zx+xyz

Using Cofactors of elements of third column, evaluate Delta=det[[1,x,yz1,y,zx1,z,xy]]

Given that xyz = -1 , the value of the determinant |(x,x^(2),1 +x^(3)),(y,y^(2),1 + y^(3)),(z,z^(2),1 +z^(3))| is

Find the value of : |{:(1,x,yz),(1,y,zx),(1,z,xy):}|

The determinant |[ C(x,1) ,C(x,2), C(x,3)] , [C(y,1) ,C(y,2), C(y,3)] , [C(z,1) ,C(z,2), C(z,3)]|= (i) 1/3xyz(x+y)(y+z)(z+x) (ii) 1/4xyz(x+y-z)(y+z-x) (iii) 1/12xyz(x-y)(y-z)(z-x) (iv) none

Prove the following : |(1,x,x^(2)-yz),(1,y,y^(2)-zx),(1,z,z^(2)-xy)|=0 .

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Using properties of determinants prove that ((1,1, 1+3x),(1+3y,1,1),(...
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|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|=
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