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Question

If a, b, c are non-zero real numbers and if the system of equations
(a − 1) x = y + z
(b − 1) y = z + x
(c − 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.

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Solution

The three equations can be expressed as

a - 1x - y - z = 0-x + b - 1y - z = 0-x - y + c - 1z = 0

Expressing this as a determinant, we get

=a-1-1-1-1b-1-1-1-1c-1

If the matrix has a non-trivial solution, then

a-1-1-1-1b-1-1-1-1c-1=0

a - 1b - 1c - 1 - 1 + 1-c - 1 - 1 - 11 + b - 1 = 0a - 1bc - c - b + 1 - 1 + 1-c + 1 - 1 - 1b = 0a - 1bc - b - c - c - b = 0abc - ab - ac - bc + b + c - b - c = 0ab + ac + bc = abc

Hence proved.

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