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

|(a - b, b-c,c-a),(b-c,c-a,a-b),(c-a,a-b,b-c)|=0

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

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

Using the property of determinants and without expanding prove that {:|( 1,bc,a(a+c) ),(1,ca,b( c+a)) ,( 1,ab,c(a+b) )|:}=0

Using the property of determinants and without expanding prove that {:|( -a^(2) , ab,ac),( ba,-b^(2) , bc) ,( ca, cb, -c^(2)) |:} =4a^(2) b^(2) c^(2)

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)|:}

Using the property of determinants prove that {:|( 3a,-a+b,-a+c),( -b+a, 3b,-b+c) ,( -c+a,-c+b,3c) |:} = 3( a+b+c) ( ab+bc+ca)

Using the Properties of determinants, prove the following: {:|(1,1,1),(a,b,c),(bc,ca,ab)|=(a-b)(b-c)(c-a)

The value of determine |{:(a-b,b+c,a),(b-a,c+a,b),(c-a,a+b,c):}| is

|(0,a,-b),(-a,0,-c),(b,c,0)| = 0