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Using the property of determinants and w...

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

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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 properties of determinants, prove that following |(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)|=(a+b+c)^3

Without expanding prove that |{:(1,a,a^(2)-bc),(1,b,b^(2)-ca),(1,c,c^(2)-ab):}|=0

Without expanding the determinant, prove that {:|( 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)) |:}