Question

If +
+
=
+
+
then prove that = or + + + = 0.​

Answer 1

Step-by-step explanation:

If a+b+c ≠ 0 and |a ,b, c, b, c, a, c, a, b| = 0, then prove that a = b = c.

Answer 2

Answer:

Answer Text

Solution :

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

`R_1 -> R_1 + R_2 + R_3`

`|(a+b+c, a+b+c, a+b+c),(b,c,a),(c,a,b)| = 0`

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

`C_2-> C_2 - C_1 & C_3 -> C_3 - C_1`

`(a+b+c)|(1,0,0),(b,c-b,a-b),(c,a-c,b-c)| = 0`

`(a+b+c)[(c-b)(b-c) - (a-b)(a-c)] = 0`

`(a+b+c)[bc-b^2 - c^2 + bc - a^2 + ab + bc + ac] = 0`

`(a+b+c)[ -a^2 - b^2 - c^2 + bc + ab + ac] = 0`

`-1/2(a+b+c)(a^2 + b^2 - 2ab + a^2 + c^2 - 2ac + b^2 + c^2 - 2bc) = 0`

`=-1/2(a+b+c)[(a-b)^2 + (b-c)^2 + (a-c)^2]= 0`

As `a+b+c != 0`

`:. (a-b)^2 + (b-c)^2 + (a-c)^2 = 0`

`a-b = 0 => a=b`

`b-c = 0 => b=c `

`c-a = 0 => c=a`

`a=b=c`

hence proved