`|{:(1,a,b+c),(1,b,c+a),(1,c,a+b):}|=0`

Prove: |(1,a, b c),(1,b ,c a),(1,c ,a b)|=|(1,a ,a^2),( 1,b,b^2),( 1,c,c^2)|

|{:(b+c,a,1),(c+a,b,1),(a+b,c,1):}|=0

|{:(a-b,1,a),(b-c,1,b),(c-a,1,c):}|=|{:(a,1,b),(b,1,c),(c,1,a):}|

Show without expanding at any stage that: [1,a,b+c],[1,b,c+a],[1,c,a+b]|=0

|{:(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b)):}|=0

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b)):}|=0

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

|{:(-1,b,c),(a,-1,c),(a,b,-1):}|=(a+1)(b+1)(c+1)((a)/(a+1)+(b)/(b+1)+(c)/(c+1)-1)

|(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b))|= 0

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