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Using the property of determinants and w...

Using the property of determinants and without expanding `{:[( x,a,x+a),( y,b,y+b),(z,c,z+c)]:} =0 `

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Using the property of determinants and without expanding {:[( 0,a,-b),(-a,0,-c),( b,c,0) ]:}=0

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,p,x),( b,q,y),(c,r,z)]:}

Knowledge Check

  • if a, b,c are non zero real numbers and if the equations (a-1)x = y+z, (b-1)y = z+x, (c-1)z = x+y have a non - trivial solution , then ab + bc + ca =

    A
    a+b+c
    B
    a b c
    C
    0
    D
    `a^2b^2c^2`

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    By using properties of determinants , show that : {:[( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) ]:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

    Without expanding prove that Delta ={:[( x+y,y+z,z+x) ,( z,x,y),( 1,1,1) ]:} =0

    Show that |{:(y+z, x, x),(y, z+x, y),(z, z, x+y):}|= 4xyz

    Prove that |{:(y+z,x,x),(y,z+x,y),(z,z,x+y):}|=4xyz

    Without expanding the determinant, prove that (i) |{:(a,a^(2),bc),(b,b^(2),ca),(c,c^(2),ab):}|=|{:(1,a^(2),a^(3)),(1,b^(2),b^(3)),(1,c^(2),c^(3)):}| (ii) |{:(ax,by,cz),(x^(2),y^(2),z^(2)),(1,1,1):}|=|{:(a,b,c),(x,y,z),(yz,zx,xy):}| (iii) |{:(1,bc,b+c),(1,ca,c+a),(1,ab,a+b):}|=|{:(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2)):}|

    Without expanding the determinant , prove that |{:(ax,by,cz),(x^2,y^2,z^2),(1,1,1):}|=|{:(a,b,c),(x,y,z),(yz,zx,xy):}|

    [(x,y,z)][(a,h,g),(h,b,f),(g,f,c)][(x),(y),(z)]=

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