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If The System Of Linear Equations a(y+z)−x=0,b(z+x)−y=0,c(x+y)−z=0 Has a Non Trivial Solution

Referencing Styles : MLA | Pages : 2

a(y+z) – x = 0 => -x + ay + az = 0 => x – ay – az = 0

b(z+x) – y = 0 => bx – y + bz = 0

c(x+y) – z = 0 => cx + cy – z = 0

Hence, the above equations are homogeneous and has non-zero trivial solutions then


 

Hence,  = 0

ð 1(1-bc)  +a(-b-bc) –a(bc + c) = 0

ð 1 – bc - ab - abc – abc - ac = 0

ð 1 – bc – ab – ac = 2abc

ð 1 = ab + bc + ac + 2abc (1)

Now, it is required to be proved

 

1 = ab + bc + ac + 2abc

Hence, this gives the result same as equation (1) and hence the question is proved.

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