Without expanding at any state, prove that `|(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b))|=0`

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

Without expanding at any state, prove that |((1)/(a),a,bc),((1)/(b),b,ca),((1)/(c ),c,ab)|=0

Without expanding at any stage, prove that |(a-b,1,a),(b-c,1,b),(c-a,1,c)| = |(a,1,b),(b,1,c),(c,1,a)|

Using the property of determinants and without expanding, prove that: |[1,b c, a(b+c)],[1,c a, b(c+a)],[1,a b, c(a+b)]|=0

witout expanding at any stage Prove that |{:(0,,a,,-b),(-a,,0,,-c),(b,,c,,0):}| =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)|

Using the property of determinants and without expanding, prove that: |1b c a(b+c)1c a b(c+a)1a b x(a+b)|=0

Using the property of determinants and without expanding in questions 1 to 7 prove that , |{:(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 in questions 1 to 7 prove that , |{:(x,a,x+a),(y,b,y+b),(z,c,z+c):}|=0

Show without expanding at any stage that: |[a+b,b+c,c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]|=2|[a,b,c],[b,c,a],[c,a,b]|